Applications of Permutations and Combinations

Question 1

The number of ways to arrange 5 different books in a row is:

Accepted Answer

120 — 5! = 5 × 4 × 3 × 2 × 1 = 120.

Question 2

The number of ways to choose a committee of 3 from 8 people is:

Accepted Answer

56 — ⁸C₃ = 8!/(3!5!) = 56 (order does not matter).

Question 3

ⁿPᵣ counts the number of:

Accepted Answer

ordered arrangements of r from n — ⁿPᵣ = n!/(n − r)! is the number of ordered arrangements.

Question 4

The number of ways to pick 2 from 10 when order matters is:

Accepted Answer

90 — ¹⁰P₂ = 10 × 9 = 90.

Question 5

How many 3-letter codes with no repeats can be formed from 5 distinct letters?

Accepted Answer

60 — ⁵P₃ = 5 × 4 × 3 = 60 (order matters, no repetition).

Question 6

The number of ways to choose 4 from 6 is:

Accepted Answer

15 — ⁶C₄ = 6!/(4!2!) = 15.

Question 7

The number of ways to arrange 4 people from 7 in a row is:

Accepted Answer

840 — ⁷P₄ = 7 × 6 × 5 × 4 = 840.

Question 8

From 7 men and 4 women, a team of 5 with exactly 3 men can be chosen in:

Accepted Answer

210 ways — ⁷C₃ × ⁴C₂ = 35 × 6 = 210.

Question 9

The number of ways to choose a president and a secretary from 12 people is:

Accepted Answer

132 — ¹²P₂ = 12 × 11 = 132 (the two roles are distinct, so order matters).

Question 10

How many handshakes occur if 6 people each shake hands once?

Accepted Answer

15 — ⁶C₂ = 6!/(2!4!) = 15 (each pair shakes once, order irrelevant).