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 ‌6C0=‌
6!
0!(6−0)!
=‌
1
1
=1 ⇒ Number of ways to select 0 items =1 For r=1 ‌6C1=‌
6!
1!(6−1)!
=‌
6
1
=6 ⇒ Number of ways to select 1 item =6 For r=2 ‌6C2=‌
6!
2!(6−2)!
=‌
6×5
2×1
=15 ⇒ Number of ways to select 2 items =15 For r=3 ‌6C3=‌
6!
3!(6−3)!
=‌
6×5×4
3×2×1
=20 ⇒ Number of ways to select 3 items = 20
Total number of selections: ‌6C0+‌6C1+‌6C2+‌6C3=1+6+15+20=42 ∴ The total number of selections of at most 3 things from 6 different things is 42 .