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Handshake Problems with Constraints - Intermediate Level: tricky scenarios handling Handshake Problems with Constraints INTERMEDIATE

This expert challenge 📈 worksheet focuses on Handshake Problems with Constraints - a key topic in Permutation Combination. You'll solve 20 intermediate-level problems (Worksheet 5 of 10). The primary focus is on tricky scenarios handling. Master how to solve handshake problems with constraints, handshake problems with constraints tricks, and handshake problems with constraints shortcut methods through systematic practice.

📝 Worksheet 5 of 10 • 20 questions • ⏱️ Estimated time: 20 minutes • 🎯 Intermediate level

What you'll learn in this worksheet:
Your progress through Handshake Problems with Constraints
Worksheet 5 of 10 (44% complete)

Question 1

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 2

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 3

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 4

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 5

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 6

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 7

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 8

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 9

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 10

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 11

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 12

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 13

In a group of 8 people, 4 people are VIPs who shake hands with everyone. The remaining 4 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: 4 (shake with everyone)
- Non-VIPs: 4 (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 = 4 × 4 = 16

Step 3 - Subtract restricted handshakes:
Actual handshakes = Total possible - Forbidden handshakes
= 28 - 16
= 12

Alternative direct count:
- VIP to VIP: C(4, 2) = 6
- VIP to Non-VIP: 0 (forbidden)
- Non-VIP to Non-VIP: C(4, 2) = 6
Total = 6 + 6 = 12

Verification: Both methods give the same result.

Question 14

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 15

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 16

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 17

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 18

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 19

At a party of 12 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: 12
- Non-participants: 3
- Participants: 9

Step 1 - Identify participants:
Only 9 people actually shake hands.

Step 2 - Calculate handshakes among participants:
Handshakes = C(9, 2) = 9×8/2

Calculation:
= 9×8/2 = 36
= 36

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

Question 20

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
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