Handshake Problems with Constraints | Worksheet 7/10

Question 1

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 2

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 3

At a party of 13 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: 13
- Non-participants: 2
- 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 4

At a party of 12 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: 12
- Non-participants: 2
- 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 5

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 6

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 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 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 9

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 10

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 11

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 12

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 13

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 14

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 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

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 17

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 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 13 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: 13
- Non-participants: 4
- 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 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

Master Handshake Problems with Constraints - Intermediate-Advanced Level Problems Handshake Problems with Constraints INTERMEDIATE ADVANCED

What you'll learn in this worksheet:

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20