Master Selection with Mandatory Constraint - Intermediate-Advanced Level Problems Selection with Mandatory Constraint INTERMEDIATE ADVANCED

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 12
- Committee size: 6
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 6 - 2 = 4
- People available for remaining positions: 12 - 2 = 10

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 4 people from remaining 10 people:
C(10,4) = 210

Calculation:
C(10,4) = (10)! / [4! × (6)!]
= 3628800 / [24 × 720]
= 210

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 4 more from 10 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 12 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(12,6) - C(12-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 210 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 10
- Committee size: 5
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 5 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 5 - 2 = 3
- People available for remaining positions: 10 - 2 = 8

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 8 people:
C(8,3) = 56

Calculation:
C(8,3) = (8)! / [3! × (5)!]
= 40320 / [6 × 120]
= 56

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 3 more from 8 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 5 from remaining 10 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(10,5) - C(10-1,5)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,5-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 56 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 11
- Committee size: 5
- Must include: 3 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 5 people total, with 3 already fixed.
- Fixed positions: 3 (these specific people are already in)
- Remaining positions to fill: 5 - 3 = 2
- People available for remaining positions: 11 - 3 = 8

Step 1 - Fix Mandatory Members:
3 specific people must be included: C(3,3) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 2 people from remaining 8 people:
C(8,2) = 28

Calculation:
C(8,2) = (8)! / [2! × (6)!]
= 40320 / [2 × 720]
= 28

Alternative Approach - Verification:
Think of it as: "We've used 3 spots, now choose 2 more from 8 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 5 from remaining 11 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(11,5) - C(11-1,5)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,5-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 28 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 12
- Committee size: 5
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 5 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 5 - 2 = 3
- People available for remaining positions: 12 - 2 = 10

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 10 people:
C(10,3) = 120

Calculation:
C(10,3) = (10)! / [3! × (7)!]
= 3628800 / [6 × 5040]
= 120

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 3 more from 10 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 5 from remaining 12 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(12,5) - C(12-1,5)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,5-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 120 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 12
- Committee size: 6
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 6 - 2 = 4
- People available for remaining positions: 12 - 2 = 10

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 4 people from remaining 10 people:
C(10,4) = 210

Calculation:
C(10,4) = (10)! / [4! × (6)!]
= 3628800 / [24 × 720]
= 210

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 4 more from 10 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 12 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(12,6) - C(12-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 210 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 12
- Committee size: 6
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 6 - 2 = 4
- People available for remaining positions: 12 - 2 = 10

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 4 people from remaining 10 people:
C(10,4) = 210

Calculation:
C(10,4) = (10)! / [4! × (6)!]
= 3628800 / [24 × 720]
= 210

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 4 more from 10 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 12 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(12,6) - C(12-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 210 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 10
- Committee size: 5
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 5 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 5 - 2 = 3
- People available for remaining positions: 10 - 2 = 8

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 8 people:
C(8,3) = 56

Calculation:
C(8,3) = (8)! / [3! × (5)!]
= 40320 / [6 × 120]
= 56

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 3 more from 8 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 5 from remaining 10 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(10,5) - C(10-1,5)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,5-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 56 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 11
- Committee size: 6
- Must include: 3 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 3 already fixed.
- Fixed positions: 3 (these specific people are already in)
- Remaining positions to fill: 6 - 3 = 3
- People available for remaining positions: 11 - 3 = 8

Step 1 - Fix Mandatory Members:
3 specific people must be included: C(3,3) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 8 people:
C(8,3) = 56

Calculation:
C(8,3) = (8)! / [3! × (5)!]
= 40320 / [6 × 120]
= 56

Alternative Approach - Verification:
Think of it as: "We've used 3 spots, now choose 3 more from 8 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 11 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(11,6) - C(11-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 56 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 12
- Committee size: 6
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 6 - 2 = 4
- People available for remaining positions: 12 - 2 = 10

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 4 people from remaining 10 people:
C(10,4) = 210

Calculation:
C(10,4) = (10)! / [4! × (6)!]
= 3628800 / [24 × 720]
= 210

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 4 more from 10 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 12 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(12,6) - C(12-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 210 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 12
- Committee size: 6
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 6 - 2 = 4
- People available for remaining positions: 12 - 2 = 10

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 4 people from remaining 10 people:
C(10,4) = 210

Calculation:
C(10,4) = (10)! / [4! × (6)!]
= 3628800 / [24 × 720]
= 210

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 4 more from 10 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 12 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(12,6) - C(12-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 210 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 11
- Committee size: 6
- Must include: 3 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 3 already fixed.
- Fixed positions: 3 (these specific people are already in)
- Remaining positions to fill: 6 - 3 = 3
- People available for remaining positions: 11 - 3 = 8

Step 1 - Fix Mandatory Members:
3 specific people must be included: C(3,3) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 8 people:
C(8,3) = 56

Calculation:
C(8,3) = (8)! / [3! × (5)!]
= 40320 / [6 × 120]
= 56

Alternative Approach - Verification:
Think of it as: "We've used 3 spots, now choose 3 more from 8 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 11 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(11,6) - C(11-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 56 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 11
- Committee size: 6
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 6 - 2 = 4
- People available for remaining positions: 11 - 2 = 9

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 4 people from remaining 9 people:
C(9,4) = 126

Calculation:
C(9,4) = (9)! / [4! × (5)!]
= 362880 / [24 × 120]
= 126

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 4 more from 9 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 11 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(11,6) - C(11-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 126 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 10
- Committee size: 5
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 5 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 5 - 2 = 3
- People available for remaining positions: 10 - 2 = 8

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 8 people:
C(8,3) = 56

Calculation:
C(8,3) = (8)! / [3! × (5)!]
= 40320 / [6 × 120]
= 56

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 3 more from 8 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 5 from remaining 10 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(10,5) - C(10-1,5)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,5-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 56 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 12
- Committee size: 5
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 5 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 5 - 2 = 3
- People available for remaining positions: 12 - 2 = 10

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 10 people:
C(10,3) = 120

Calculation:
C(10,3) = (10)! / [3! × (7)!]
= 3628800 / [6 × 5040]
= 120

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 3 more from 10 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 5 from remaining 12 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(12,5) - C(12-1,5)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,5-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 120 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 12
- Committee size: 5
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 5 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 5 - 2 = 3
- People available for remaining positions: 12 - 2 = 10

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 10 people:
C(10,3) = 120

Calculation:
C(10,3) = (10)! / [3! × (7)!]
= 3628800 / [6 × 5040]
= 120

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 3 more from 10 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 5 from remaining 12 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(12,5) - C(12-1,5)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,5-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 120 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 10
- Committee size: 6
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 6 - 2 = 4
- People available for remaining positions: 10 - 2 = 8

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 4 people from remaining 8 people:
C(8,4) = 70

Calculation:
C(8,4) = (8)! / [4! × (4)!]
= 40320 / [24 × 24]
= 70

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 4 more from 8 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 10 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(10,6) - C(10-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 70 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 11
- Committee size: 6
- Must include: 3 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 3 already fixed.
- Fixed positions: 3 (these specific people are already in)
- Remaining positions to fill: 6 - 3 = 3
- People available for remaining positions: 11 - 3 = 8

Step 1 - Fix Mandatory Members:
3 specific people must be included: C(3,3) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 8 people:
C(8,3) = 56

Calculation:
C(8,3) = (8)! / [3! × (5)!]
= 40320 / [6 × 120]
= 56

Alternative Approach - Verification:
Think of it as: "We've used 3 spots, now choose 3 more from 8 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 11 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(11,6) - C(11-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 56 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 10
- Committee size: 5
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 5 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 5 - 2 = 3
- People available for remaining positions: 10 - 2 = 8

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 8 people:
C(8,3) = 56

Calculation:
C(8,3) = (8)! / [3! × (5)!]
= 40320 / [6 × 120]
= 56

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 3 more from 8 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 5 from remaining 10 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(10,5) - C(10-1,5)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,5-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 56 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 12
- Committee size: 6
- Must include: 3 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 6 people total, with 3 already fixed.
- Fixed positions: 3 (these specific people are already in)
- Remaining positions to fill: 6 - 3 = 3
- People available for remaining positions: 12 - 3 = 9

Step 1 - Fix Mandatory Members:
3 specific people must be included: C(3,3) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 9 people:
C(9,3) = 84

Calculation:
C(9,3) = (9)! / [3! × (6)!]
= 362880 / [6 × 720]
= 84

Alternative Approach - Verification:
Think of it as: "We've used 3 spots, now choose 3 more from 9 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 6 from remaining 12 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(12,6) - C(12-1,6)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,6-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 84 ways

Step-by-Step Solution:

Concept: Combination with mandatory inclusion constraint.

Given:
- Total people: 11
- Committee size: 5
- Must include: 2 specific people

Strategy: Fix the mandatory selections first, then choose remaining from available pool.

Analysis:
We need to select 5 people total, with 2 already fixed.
- Fixed positions: 2 (these specific people are already in)
- Remaining positions to fill: 5 - 2 = 3
- People available for remaining positions: 11 - 2 = 9

Step 1 - Fix Mandatory Members:
2 specific people must be included: C(2,2) = 1 way
(This is automatic - we have no choice here)

Step 2 - Select Remaining Members:
Choose 3 people from remaining 9 people:
C(9,3) = 84

Calculation:
C(9,3) = (9)! / [3! × (6)!]
= 362880 / [6 × 720]
= 84

Alternative Approach - Verification:
Think of it as: "We've used 2 spots, now choose 3 more from 9 remaining"

Related Problem Types:

1. Must EXCLUDE specific people:
Select all 5 from remaining 11 - (people to exclude)

2. At least one specific person:
Total ways - Ways without that person
= C(11,5) - C(11-1,5)

3. Exactly k from group A, rest from group B:
C(|A|,k) × C(|B|,5-k)

Common Error: Don't forget to reduce both the total pool and the selection size by the number of mandatory inclusions.

Answer: 84 ways