Question

Suppose 6 of your friends have asked to go to the latest sci-fi movie with you. You have studied your calendar and decided to see it on three particular occasions before it leaves town. If you plan to invite each of your 6 friends to join you once, how many ways can you do so? Suppose you want to have 2 friends with you each time you see it, how many ways can you do this?

Answer 1

a) To invite each of 6 friends to join him once he can invite one or more friends by inviting 1 friend, or 2 friends or 3 friend or 4 or 5 friends or  all the 6 friends.

1 friend can be selected out of 6 in ⁶C₁ = 6 ways

2 friends can be selected out of 6 in ⁶C₂ = 15 ways

3 friends can be selected out of 6 in ⁶C₃ = 20 ways

4 friends can be selected out of 6 in ⁶C₄ = 15 ways

5 friends can be selected out of 6 in ⁶C₅ = 6 ways

6 friends can be selected out of 6 in ⁶C₆ = 1 ways

The required number of ways (combinations) for one occasion = 6 + 15 + 20 + 15 + 6 + 1 = 63

Therefore the required number of ways (combinations) for all three occasions = 3*63 = 189

b) To have 2 friends with him each time he can invite 2 friends out of 6 in ⁶C₂ ways in one occasion.

Therefore the required number of ways (combinations) for all three occasions = 3*⁶C₃ = 60