In how many ways at least two team members can be selected for grade A out of…

Answer: 7C2 + 7C3 + 7C4 + 7C5 + 7C6 + 7C7 = 120

Explanation: At least two means choosing 2 or more members from the 7, so sum the combinations for k = 2 to 7.

• Number of ways to select 2 members: 7C2 = 21

• Number of ways to select 3 members: 7C3 = 35

• Number of ways to select 4 members: 7C4 = 35

• Number of ways to select 5 members: 7C5 = 21

• Number of ways to select 6 members: 7C6 = 7

• Number of ways to select 7 members: 7C7 = 1

Total (sum): 21 + 35 + 35 + 21 + 7 + 1 = 120

Quick method: Sum from k = 2 to 7 of 7Ck = 2^7 − C(7,0) − C(7,1) = 128 − 1 − 7 = 120