What is the number of selections of at most 3 things from 6 different things?

Calculation:

Given:

Total number of items, n = 6

We need to calculate the total number of selections for r = 0, 1, 2, and 3:

For r = 0

\( ^6C_0 = \frac{6!}{0!(6-0)!} = \frac{1}{1} = 1 \)

⇒ Number of ways to select 0 items = 1

For r = 1

\( ^6C_1 = \frac{6!}{1!(6-1)!} = \frac{6}{1} = 6 \)

⇒ Number of ways to select 1 item = 6

For r = 2

\( ^6C_2 = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \)

⇒ Number of ways to select 2 items = 15

For r = 3

\( ^6C_3 = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \)

⇒ Number of ways to select 3 items = 20

Total number of selections:

\( ^6C_0 + ^6C_1 + ^6C_2 + ^6C_3 = 1 + 6 + 15 + 20 = 42 \)

∴ The total number of selections of at most 3 things from 6 different things is 42.