The notation n!!
I am a little confused about the n!! notation.
If n is even, do we do (n-1)!!? In the video there are n=8 people, and then we subtract one to get 1 to get 7 (n-1), which we then do 7!! to get 105.
How about if n is odd? It may seem weird to pair up odd numbers of people, but what if one group could have 3 people or the person that is not picked is simply left out?
For example, if n is 9, would we do n-1 = 8 which leads to 8!! = 8 * 6 * 4 * 2?
@西瓜 To answer your questions:
You are right that 8!! = 8 * 6 * 4 * 2. However, you can't really split 9 people into pairs of two, so you can't really use 8!! for 9 people. You can still use the idea of double factorial though!
If you wanted to find the number of ways to split the 9 people into 4 pairs and 1 person by themself, you could first choose the person who is by themself (9 ways to do this) and then multiply by the number of ways to split the 8 remaining people into four pairs (7!!) so the answer would be 9 * 7!!
If you wanted to split the 9 people into 3 pairs and 1 group of three, you could first choose the group of 3 (9 choose 3 ways to do this) and then multiply by the number of ways to split the 6 remaining people into three pairs (5!!) so the answer would be (9 choose 3) * 5!!
@v4913 I understand now. Thank you very much!