# Counting number of groups

Find the number of ways of forming a group of $2k$ people from $n$ couples,where $n,k \in \mathbb{N}$ with $2k \le n$, in each of the following cases: (i) There are $k$ couples in such a group; (ii) No couples are included in such a group: (iii) At least one couple is in included in such a group; (iv) Exactly two couples are included in such a group.

• Hint for (ii): first choose the $2k$ couples. Then choose one member of each selected couple. – lulu Apr 21 '17 at 13:26
• For case (ii), you can first select $2k$ couples and then select one person out of each couple. As such, the answer is ${{n}\choose{2k}} 2^{2k}$. For case (iv), first select 2 couples of which you will select both partners, then select $2k-4$ couples of which you will select one person. The total number of combinations is then ${{n}\choose{2}} {{n-2}\choose{2k-4}} 2^{2k-4}$. – jvdhooft May 3 '17 at 11:13