markp@tekmdp.UUCP (Mark Paulin) (09/14/83)
In how many ways may 2r people be chosen from n married couples in such a way
that we have exactly k married couples among the 2r people? k <= r <= n.
1) Let C(a, b) denote the binomial coefficient "a above b". Then first choose
k couples from the n couples -- this may be done in C(k, n) ways.
2) Next choose, from the remaining (n - k) couples, the 2(r - k) couples which
will be represented among the 2r people to be assembled. This may be done
in C(2(r - k), (n - k)) ways.
3) Finally, for each of the 2(r - k) couples chosen in 2), choose the member to
be included among the 2r people. This may be done in 2 ways per couple, or
a total of 2**(2(r - k)) ways.
Thus the total number of ways to form the required group of 2r people is:
C(n, k) * C(2(r - k), (n - k)) * 2**(2(r - k))
///.
Mark Paulin
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