gistfile1.txt

Q: i want to choose 4 people out of 4 couples. i want a group with at least 1 couple.

4 couples:
  ab
  cd
  ef
  gh

> > > selecting among the four couples is:
> > > (4 choose 1) = 4


> > > This part of your work was the AHA! for me to see the formal answer:

non-couples:
  6:a(c,d,e,f,g,h)
  6:b(c,d,e,f,g,h)
  4:c(e,f,g,h)
  4:d(e,f,g,h)
  2:e(g,h)
  2:f(g,h)
  --
  24

  pre-selecting one couple reduces # of selectable non-couples to 12.

> > > 12 is the number of non-couples after pre-selection:
> > > (3 choose 2) * (2 choose 1) * (2 choose 1) = 12
> > > choose two of remaining couples, and from each couple, choose 1 person

couple(couple):
  3:ab(cd,ef,gh)
  2:cd(ef,gh)
  1:ef(gh)
  -
  6

> > > this is the two couple scenario:
> > > (4 choose 2) = 6

for each couple:
  couple(12 non-couples + remaining selectable couples)

15:ab(12 + 3)
14:cd(12 + 2)
13:ef(12 + 1)
12:gh(12)
--
54
==

> > > the possible ways to pre-select is (4 choose 1) = 4, and thus
> > > the number of ways to pre-select a couple, and then choose a non-couple is:
> > > 4 * 12 = 48

> > > And then add that to the ways to select 2 couples:
> > > 6 + 48 = 54