Committee Formation - Intermediate Level: tricky scenarios handling Committee Formation INTERMEDIATE

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 5
- Committee size: 4
- Required: 2 men and 2 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 2 men from 8 men = C(8,2)
C(8,2) = 8! / [2! × 6!] = 28

Step 2 - Select Women:
Choose 2 women from 5 women = C(5,2)
C(5,2) = 5! / [2! × 3!] = 10

Step 3 - Apply Multiplication Principle:
Total ways = C(8,2) × C(5,2)
= 28 × 10
= 280

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 4
- Committee size: 5
- Required: 3 men and 2 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 3 men from 8 men = C(8,3)
C(8,3) = 8! / [3! × 5!] = 56

Step 2 - Select Women:
Choose 2 women from 4 women = C(4,2)
C(4,2) = 4! / [2! × 2!] = 6

Step 3 - Apply Multiplication Principle:
Total ways = C(8,3) × C(4,2)
= 56 × 6
= 336

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 5
- Committee size: 6
- Required: 3 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 3 men from 8 men = C(8,3)
C(8,3) = 8! / [3! × 5!] = 56

Step 2 - Select Women:
Choose 3 women from 5 women = C(5,3)
C(5,3) = 5! / [3! × 2!] = 10

Step 3 - Apply Multiplication Principle:
Total ways = C(8,3) × C(5,3)
= 56 × 10
= 560

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 7
- Women available: 4
- Committee size: 6
- Required: 3 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 3 men from 7 men = C(7,3)
C(7,3) = 7! / [3! × 4!] = 35

Step 2 - Select Women:
Choose 3 women from 4 women = C(4,3)
C(4,3) = 4! / [3! × 1!] = 4

Step 3 - Apply Multiplication Principle:
Total ways = C(7,3) × C(4,3)
= 35 × 4
= 140

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 6
- Committee size: 4
- Required: 2 men and 2 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 2 men from 8 men = C(8,2)
C(8,2) = 8! / [2! × 6!] = 28

Step 2 - Select Women:
Choose 2 women from 6 women = C(6,2)
C(6,2) = 6! / [2! × 4!] = 15

Step 3 - Apply Multiplication Principle:
Total ways = C(8,2) × C(6,2)
= 28 × 15
= 420

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 5
- Committee size: 4
- Required: 2 men and 2 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 2 men from 8 men = C(8,2)
C(8,2) = 8! / [2! × 6!] = 28

Step 2 - Select Women:
Choose 2 women from 5 women = C(5,2)
C(5,2) = 5! / [2! × 3!] = 10

Step 3 - Apply Multiplication Principle:
Total ways = C(8,2) × C(5,2)
= 28 × 10
= 280

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 5
- Committee size: 6
- Required: 3 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 3 men from 8 men = C(8,3)
C(8,3) = 8! / [3! × 5!] = 56

Step 2 - Select Women:
Choose 3 women from 5 women = C(5,3)
C(5,3) = 5! / [3! × 2!] = 10

Step 3 - Apply Multiplication Principle:
Total ways = C(8,3) × C(5,3)
= 56 × 10
= 560

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 7
- Women available: 5
- Committee size: 4
- Required: 2 men and 2 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 2 men from 7 men = C(7,2)
C(7,2) = 7! / [2! × 5!] = 21

Step 2 - Select Women:
Choose 2 women from 5 women = C(5,2)
C(5,2) = 5! / [2! × 3!] = 10

Step 3 - Apply Multiplication Principle:
Total ways = C(7,2) × C(5,2)
= 21 × 10
= 210

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 5
- Committee size: 6
- Required: 3 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 3 men from 8 men = C(8,3)
C(8,3) = 8! / [3! × 5!] = 56

Step 2 - Select Women:
Choose 3 women from 5 women = C(5,3)
C(5,3) = 5! / [3! × 2!] = 10

Step 3 - Apply Multiplication Principle:
Total ways = C(8,3) × C(5,3)
= 56 × 10
= 560

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 7
- Women available: 6
- Committee size: 5
- Required: 3 men and 2 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 3 men from 7 men = C(7,3)
C(7,3) = 7! / [3! × 4!] = 35

Step 2 - Select Women:
Choose 2 women from 6 women = C(6,2)
C(6,2) = 6! / [2! × 4!] = 15

Step 3 - Apply Multiplication Principle:
Total ways = C(7,3) × C(6,2)
= 35 × 15
= 525

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 6
- Committee size: 6
- Required: 3 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 3 men from 8 men = C(8,3)
C(8,3) = 8! / [3! × 5!] = 56

Step 2 - Select Women:
Choose 3 women from 6 women = C(6,3)
C(6,3) = 6! / [3! × 3!] = 20

Step 3 - Apply Multiplication Principle:
Total ways = C(8,3) × C(6,3)
= 56 × 20
= 1120

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 6
- Committee size: 5
- Required: 2 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 2 men from 8 men = C(8,2)
C(8,2) = 8! / [2! × 6!] = 28

Step 2 - Select Women:
Choose 3 women from 6 women = C(6,3)
C(6,3) = 6! / [3! × 3!] = 20

Step 3 - Apply Multiplication Principle:
Total ways = C(8,2) × C(6,3)
= 28 × 20
= 560

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 7
- Women available: 5
- Committee size: 4
- Required: 2 men and 2 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 2 men from 7 men = C(7,2)
C(7,2) = 7! / [2! × 5!] = 21

Step 2 - Select Women:
Choose 2 women from 5 women = C(5,2)
C(5,2) = 5! / [2! × 3!] = 10

Step 3 - Apply Multiplication Principle:
Total ways = C(7,2) × C(5,2)
= 21 × 10
= 210

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 6
- Women available: 4
- Committee size: 5
- Required: 3 men and 2 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 3 men from 6 men = C(6,3)
C(6,3) = 6! / [3! × 3!] = 20

Step 2 - Select Women:
Choose 2 women from 4 women = C(4,2)
C(4,2) = 4! / [2! × 2!] = 6

Step 3 - Apply Multiplication Principle:
Total ways = C(6,3) × C(4,2)
= 20 × 6
= 120

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 7
- Women available: 4
- Committee size: 5
- Required: 2 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 2 men from 7 men = C(7,2)
C(7,2) = 7! / [2! × 5!] = 21

Step 2 - Select Women:
Choose 3 women from 4 women = C(4,3)
C(4,3) = 4! / [3! × 1!] = 4

Step 3 - Apply Multiplication Principle:
Total ways = C(7,2) × C(4,3)
= 21 × 4
= 84

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 7
- Women available: 5
- Committee size: 5
- Required: 2 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 2 men from 7 men = C(7,2)
C(7,2) = 7! / [2! × 5!] = 21

Step 2 - Select Women:
Choose 3 women from 5 women = C(5,3)
C(5,3) = 5! / [3! × 2!] = 10

Step 3 - Apply Multiplication Principle:
Total ways = C(7,2) × C(5,3)
= 21 × 10
= 210

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 4
- Committee size: 4
- Required: 2 men and 2 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 2 men from 8 men = C(8,2)
C(8,2) = 8! / [2! × 6!] = 28

Step 2 - Select Women:
Choose 2 women from 4 women = C(4,2)
C(4,2) = 4! / [2! × 2!] = 6

Step 3 - Apply Multiplication Principle:
Total ways = C(8,2) × C(4,2)
= 28 × 6
= 168

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 7
- Women available: 6
- Committee size: 6
- Required: 3 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 3 men from 7 men = C(7,3)
C(7,3) = 7! / [3! × 4!] = 35

Step 2 - Select Women:
Choose 3 women from 6 women = C(6,3)
C(6,3) = 6! / [3! × 3!] = 20

Step 3 - Apply Multiplication Principle:
Total ways = C(7,3) × C(6,3)
= 35 × 20
= 700

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 7
- Women available: 4
- Committee size: 5
- Required: 2 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 2 men from 7 men = C(7,2)
C(7,2) = 7! / [2! × 5!] = 21

Step 2 - Select Women:
Choose 3 women from 4 women = C(4,3)
C(4,3) = 4! / [3! × 1!] = 4

Step 3 - Apply Multiplication Principle:
Total ways = C(7,2) × C(4,3)
= 21 × 4
= 84

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.

Step-by-Step Solution:

Concept: Combination with constraints - selection from multiple groups with specific requirements.

Given:
- Men available: 8
- Women available: 5
- Committee size: 6
- Required: 3 men and 3 women

Strategy: Select from each group independently, then multiply (Multiplication Principle).

Step 1 - Select Men:
Choose 3 men from 8 men = C(8,3)
C(8,3) = 8! / [3! × 5!] = 56

Step 2 - Select Women:
Choose 3 women from 5 women = C(5,3)
C(5,3) = 5! / [3! × 2!] = 10

Step 3 - Apply Multiplication Principle:
Total ways = C(8,3) × C(5,3)
= 56 × 10
= 560

Key Principle: When selecting from different independent groups with specific requirements from each:
- Calculate selections from each group separately
- Multiply the results

Common Error: Don't add the combinations - multiply them! Each selection from one group can be paired with each selection from the other.