ways to choose question

Question

Jason wants to invite three of her friends from four married couple to play badminton with her.

How many ways can she invite them if

 

a) one particular couple will not attend separately.

 

b)two of them are not comfortable with each other and will not attend together.

 

c)none of the three friends she invited are husband and wife.

 

Answer 1

We have 4 married couples → 8 people. Jason invites 3 friends (i.e., 3-person subsets of the 8) under various constraints.

a) One particular couple only attends together (never separately).

Valid groups either include both or neither of that couple.

1. Exclude both: choose all 3 from the other 6 → (6⁄3) = 20.
2. Include both: choose 1 more from the other 6 → (6⁄1) = 6.
3. Total: 20 + 6 = 26.

b) Two specific people won’t attend together.

Start with all (8⁄3) = 56 groups and subtract those that contain both of the incompatible pair (then choose 1 of the remaining 6): (6⁄1) = 6.

Total: 56 − 6 = 50.

c) No invited pair are spouses.

Choose 3 distinct couples out of 4, then 1 person from each chosen couple:

(4⁄3) • 2³ = 4 • 8 = 32.

Answers:

a) 26 b) 50 c) 32