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Handshake Problems with Constraints - Absolute-Beginner Level: core concept mastery Handshake Problems with Constraints ABSOLUTE BEGINNER

This skill primer 🌟 worksheet focuses on Handshake Problems with Constraints - a key topic in Permutation Combination. You'll solve 20 absolute-beginner-level problems (Worksheet 1 of 10). The primary focus is on core concept mastery. Master handshake problems with constraints problems, handshake problems with constraints reasoning questions, and handshake problems with constraints practice through systematic practice.

📝 Worksheet 1 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Absolute Beginner level

What you'll learn in this worksheet:
Your progress through Handshake Problems with Constraints
Worksheet 1 of 10 (0% complete)

Question 1

In a group of 8 people, 3 people are VIPs who shake hands with everyone. The remaining 5 people only shake hands with other non-VIPs (not with VIPs). How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with selective participation.

Given:
- Total: 8 people
- VIPs: 3 (shake with everyone)
- Non-VIPs: 5 (only shake with other non-VIPs)

Step 1 - Total possible handshakes without restrictions:
Total possible = C(8, 2) = 8×7/2 = 28

Step 2 - Handshakes that DON'T occur:
Non-VIPs shaking with VIPs (these don't happen because non-VIPs only shake with non-VIPs)
Non-VIP to VIP handshakes = 5 × 3 = 15

Step 3 - Subtract restricted handshakes:
Actual handshakes = Total possible - Forbidden handshakes
= 28 - 15
= 13

Alternative direct count:
- VIP to VIP: C(3, 2) = 3
- VIP to Non-VIP: 0 (forbidden)
- Non-VIP to Non-VIP: C(5, 2) = 10
Total = 3 + 10 = 13

Verification: Both methods give the same result.

Question 2

In a group of 10 people, 3 people are VIPs who shake hands with everyone. The remaining 7 people only shake hands with other non-VIPs (not with VIPs). How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with selective participation.

Given:
- Total: 10 people
- VIPs: 3 (shake with everyone)
- Non-VIPs: 7 (only shake with other non-VIPs)

Step 1 - Total possible handshakes without restrictions:
Total possible = C(10, 2) = 10×9/2 = 45

Step 2 - Handshakes that DON'T occur:
Non-VIPs shaking with VIPs (these don't happen because non-VIPs only shake with non-VIPs)
Non-VIP to VIP handshakes = 7 × 3 = 21

Step 3 - Subtract restricted handshakes:
Actual handshakes = Total possible - Forbidden handshakes
= 45 - 21
= 24

Alternative direct count:
- VIP to VIP: C(3, 2) = 3
- VIP to Non-VIP: 0 (forbidden)
- Non-VIP to Non-VIP: C(7, 2) = 21
Total = 3 + 21 = 24

Verification: Both methods give the same result.

Question 3

In a group of 10 people, 3 people are VIPs who shake hands with everyone. The remaining 7 people only shake hands with other non-VIPs (not with VIPs). How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with selective participation.

Given:
- Total: 10 people
- VIPs: 3 (shake with everyone)
- Non-VIPs: 7 (only shake with other non-VIPs)

Step 1 - Total possible handshakes without restrictions:
Total possible = C(10, 2) = 10×9/2 = 45

Step 2 - Handshakes that DON'T occur:
Non-VIPs shaking with VIPs (these don't happen because non-VIPs only shake with non-VIPs)
Non-VIP to VIP handshakes = 7 × 3 = 21

Step 3 - Subtract restricted handshakes:
Actual handshakes = Total possible - Forbidden handshakes
= 45 - 21
= 24

Alternative direct count:
- VIP to VIP: C(3, 2) = 3
- VIP to Non-VIP: 0 (forbidden)
- Non-VIP to Non-VIP: C(7, 2) = 21
Total = 3 + 21 = 24

Verification: Both methods give the same result.

Question 4

There are 10 people at a party. If each person shakes hands with every other person exactly once, how many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem - each handshake involves 2 people, and order doesn't matter.

Given: 10 people

Method 1 - Combination:
Number of handshakes = Number of ways to choose 2 people from 10
= C(10, 2) = 10! / (2! × 8!) = 10×9/2

Method 2 - Counting per person:
Each person shakes hands with 9 others
Total count: 10 × 9 = 90
But each handshake counted twice, so divide by 2: 90/2

Calculation:
= 10×9/2 = 45
= 45

Key Formula: Number of handshakes = C(n, 2) = n(n-1)/2

Question 5

There are 15 people at a party. If each person shakes hands with every other person exactly once, how many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem - each handshake involves 2 people, and order doesn't matter.

Given: 15 people

Method 1 - Combination:
Number of handshakes = Number of ways to choose 2 people from 15
= C(15, 2) = 15! / (2! × 13!) = 15×14/2

Method 2 - Counting per person:
Each person shakes hands with 14 others
Total count: 15 × 14 = 210
But each handshake counted twice, so divide by 2: 210/2

Calculation:
= 15×14/2 = 105
= 105

Key Formula: Number of handshakes = C(n, 2) = n(n-1)/2

Question 6

In a group of 10 people, 4 people are VIPs who shake hands with everyone. The remaining 6 people only shake hands with other non-VIPs (not with VIPs). How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with selective participation.

Given:
- Total: 10 people
- VIPs: 4 (shake with everyone)
- Non-VIPs: 6 (only shake with other non-VIPs)

Step 1 - Total possible handshakes without restrictions:
Total possible = C(10, 2) = 10×9/2 = 45

Step 2 - Handshakes that DON'T occur:
Non-VIPs shaking with VIPs (these don't happen because non-VIPs only shake with non-VIPs)
Non-VIP to VIP handshakes = 6 × 4 = 24

Step 3 - Subtract restricted handshakes:
Actual handshakes = Total possible - Forbidden handshakes
= 45 - 24
= 21

Alternative direct count:
- VIP to VIP: C(4, 2) = 6
- VIP to Non-VIP: 0 (forbidden)
- Non-VIP to Non-VIP: C(6, 2) = 15
Total = 6 + 15 = 21

Verification: Both methods give the same result.

Question 7

At a party of 14 people, 4 people refuse to shake hands. Only the remaining people shake hands with each other exactly once. How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with subset participation.

Given:
- Total people: 14
- Non-participants: 4
- Participants: 10

Step 1 - Identify participants:
Only 10 people actually shake hands.

Step 2 - Calculate handshakes among participants:
Handshakes = C(10, 2) = 10×9/2

Calculation:
= 10×9/2 = 45
= 45

Key Point: Non-participants don't contribute to any handshake.

Question 8

At a party of 15 people, 3 people refuse to shake hands. Only the remaining people shake hands with each other exactly once. How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with subset participation.

Given:
- Total people: 15
- Non-participants: 3
- Participants: 12

Step 1 - Identify participants:
Only 12 people actually shake hands.

Step 2 - Calculate handshakes among participants:
Handshakes = C(12, 2) = 12×11/2

Calculation:
= 12×11/2 = 66
= 66

Key Point: Non-participants don't contribute to any handshake.

Question 9

There are 14 people at a party. If each person shakes hands with every other person exactly once, how many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem - each handshake involves 2 people, and order doesn't matter.

Given: 14 people

Method 1 - Combination:
Number of handshakes = Number of ways to choose 2 people from 14
= C(14, 2) = 14! / (2! × 12!) = 14×13/2

Method 2 - Counting per person:
Each person shakes hands with 13 others
Total count: 14 × 13 = 182
But each handshake counted twice, so divide by 2: 182/2

Calculation:
= 14×13/2 = 91
= 91

Key Formula: Number of handshakes = C(n, 2) = n(n-1)/2

Question 10

At a party of 15 people, 2 people refuse to shake hands. Only the remaining people shake hands with each other exactly once. How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with subset participation.

Given:
- Total people: 15
- Non-participants: 2
- Participants: 13

Step 1 - Identify participants:
Only 13 people actually shake hands.

Step 2 - Calculate handshakes among participants:
Handshakes = C(13, 2) = 13×12/2

Calculation:
= 13×12/2 = 78
= 78

Key Point: Non-participants don't contribute to any handshake.

Question 11

In a group of 10 people, 4 people are VIPs who shake hands with everyone. The remaining 6 people only shake hands with other non-VIPs (not with VIPs). How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with selective participation.

Given:
- Total: 10 people
- VIPs: 4 (shake with everyone)
- Non-VIPs: 6 (only shake with other non-VIPs)

Step 1 - Total possible handshakes without restrictions:
Total possible = C(10, 2) = 10×9/2 = 45

Step 2 - Handshakes that DON'T occur:
Non-VIPs shaking with VIPs (these don't happen because non-VIPs only shake with non-VIPs)
Non-VIP to VIP handshakes = 6 × 4 = 24

Step 3 - Subtract restricted handshakes:
Actual handshakes = Total possible - Forbidden handshakes
= 45 - 24
= 21

Alternative direct count:
- VIP to VIP: C(4, 2) = 6
- VIP to Non-VIP: 0 (forbidden)
- Non-VIP to Non-VIP: C(6, 2) = 15
Total = 6 + 15 = 21

Verification: Both methods give the same result.

Question 12

There are 13 people at a party. If each person shakes hands with every other person exactly once, how many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem - each handshake involves 2 people, and order doesn't matter.

Given: 13 people

Method 1 - Combination:
Number of handshakes = Number of ways to choose 2 people from 13
= C(13, 2) = 13! / (2! × 11!) = 13×12/2

Method 2 - Counting per person:
Each person shakes hands with 12 others
Total count: 13 × 12 = 156
But each handshake counted twice, so divide by 2: 156/2

Calculation:
= 13×12/2 = 78
= 78

Key Formula: Number of handshakes = C(n, 2) = n(n-1)/2

Question 13

At a party of 15 people, 4 people refuse to shake hands. Only the remaining people shake hands with each other exactly once. How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with subset participation.

Given:
- Total people: 15
- Non-participants: 4
- Participants: 11

Step 1 - Identify participants:
Only 11 people actually shake hands.

Step 2 - Calculate handshakes among participants:
Handshakes = C(11, 2) = 11×10/2

Calculation:
= 11×10/2 = 55
= 55

Key Point: Non-participants don't contribute to any handshake.

Question 14

There are 12 people at a party. If each person shakes hands with every other person exactly once, how many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem - each handshake involves 2 people, and order doesn't matter.

Given: 12 people

Method 1 - Combination:
Number of handshakes = Number of ways to choose 2 people from 12
= C(12, 2) = 12! / (2! × 10!) = 12×11/2

Method 2 - Counting per person:
Each person shakes hands with 11 others
Total count: 12 × 11 = 132
But each handshake counted twice, so divide by 2: 132/2

Calculation:
= 12×11/2 = 66
= 66

Key Formula: Number of handshakes = C(n, 2) = n(n-1)/2

Question 15

There are 10 people at a party. If each person shakes hands with every other person exactly once, how many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem - each handshake involves 2 people, and order doesn't matter.

Given: 10 people

Method 1 - Combination:
Number of handshakes = Number of ways to choose 2 people from 10
= C(10, 2) = 10! / (2! × 8!) = 10×9/2

Method 2 - Counting per person:
Each person shakes hands with 9 others
Total count: 10 × 9 = 90
But each handshake counted twice, so divide by 2: 90/2

Calculation:
= 10×9/2 = 45
= 45

Key Formula: Number of handshakes = C(n, 2) = n(n-1)/2

Question 16

At a party of 13 people, 3 people refuse to shake hands. Only the remaining people shake hands with each other exactly once. How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with subset participation.

Given:
- Total people: 13
- Non-participants: 3
- Participants: 10

Step 1 - Identify participants:
Only 10 people actually shake hands.

Step 2 - Calculate handshakes among participants:
Handshakes = C(10, 2) = 10×9/2

Calculation:
= 10×9/2 = 45
= 45

Key Point: Non-participants don't contribute to any handshake.

Question 17

At a party of 15 people, 3 people refuse to shake hands. Only the remaining people shake hands with each other exactly once. How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with subset participation.

Given:
- Total people: 15
- Non-participants: 3
- Participants: 12

Step 1 - Identify participants:
Only 12 people actually shake hands.

Step 2 - Calculate handshakes among participants:
Handshakes = C(12, 2) = 12×11/2

Calculation:
= 12×11/2 = 66
= 66

Key Point: Non-participants don't contribute to any handshake.

Question 18

There are 11 people at a party. If each person shakes hands with every other person exactly once, how many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem - each handshake involves 2 people, and order doesn't matter.

Given: 11 people

Method 1 - Combination:
Number of handshakes = Number of ways to choose 2 people from 11
= C(11, 2) = 11! / (2! × 9!) = 11×10/2

Method 2 - Counting per person:
Each person shakes hands with 10 others
Total count: 11 × 10 = 110
But each handshake counted twice, so divide by 2: 110/2

Calculation:
= 11×10/2 = 55
= 55

Key Formula: Number of handshakes = C(n, 2) = n(n-1)/2

Question 19

In a group of 8 people, 3 people are VIPs who shake hands with everyone. The remaining 5 people only shake hands with other non-VIPs (not with VIPs). How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with selective participation.

Given:
- Total: 8 people
- VIPs: 3 (shake with everyone)
- Non-VIPs: 5 (only shake with other non-VIPs)

Step 1 - Total possible handshakes without restrictions:
Total possible = C(8, 2) = 8×7/2 = 28

Step 2 - Handshakes that DON'T occur:
Non-VIPs shaking with VIPs (these don't happen because non-VIPs only shake with non-VIPs)
Non-VIP to VIP handshakes = 5 × 3 = 15

Step 3 - Subtract restricted handshakes:
Actual handshakes = Total possible - Forbidden handshakes
= 28 - 15
= 13

Alternative direct count:
- VIP to VIP: C(3, 2) = 3
- VIP to Non-VIP: 0 (forbidden)
- Non-VIP to Non-VIP: C(5, 2) = 10
Total = 3 + 10 = 13

Verification: Both methods give the same result.

Question 20

At a party of 15 people, 2 people refuse to shake hands. Only the remaining people shake hands with each other exactly once. How many handshakes occur?
Step-by-Step Solution:

Concept: Handshake problem with subset participation.

Given:
- Total people: 15
- Non-participants: 2
- Participants: 13

Step 1 - Identify participants:
Only 13 people actually shake hands.

Step 2 - Calculate handshakes among participants:
Handshakes = C(13, 2) = 13×12/2

Calculation:
= 13×12/2 = 78
= 78

Key Point: Non-participants don't contribute to any handshake.
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