Answer to Question #4398 in Combinatorics | Number Theory for Bliss

Question

Find the number of ways of forming a group of 2k people from n couple, where k,n are elements of N with 2k≤n in each case: i) k couples are in such a group, ii) no couples are included in such a group, iii) at least one couple is included in such a group, and iv) exactly two couples are included in such a group.

Answer 1

i) k couples are in such a groupWe should pick up k couples from n

ii) no couples are included insuch a group
at first time we can pick nvariants
second- n-2(cannot pick 1 couple one member of whitchis picked)
…
So ans n(n-2)...(n-2k)/(2k)!
iii) at least one couple isincluded in such a group
C^2k_2n-all variants to pick up 2k from 2n
n(n-2)..(n-2k)/(2k)! it does not fitus.
So ans:C^2k_2n - n(n-2)..(n-2k)/(2k)!
iv) exactly two couples are included in such a group.
C²_n-so many ways areto pick 2 couples
C^2k-4_2n-4-we pick from 2n-4people 2k-4 places in our group

so ans. C²_n*C^2k-4_2n-4