We distribute n pennies to k boys and l girls, so that (to be really unfair) we require that each girls gets at least one penny. In how many ways can we do this?
Distribute pennies for children
1
$\begingroup$
discrete-mathematics
-
0^^was it *that* hard to parse? (Also, @Nguyën, "penny" is the singular form of the plural "pennies") – 2012-06-11
1 Answers
2
I'm assuming each boy and girl is labeled (that is, giving 3 pennies to boy 1 and 5 to boy 2 is counted as distinct from giving 5 to boy 1 and 3 to boy 2).
Since each girl gets at least one penny, removing $l$ pennies and removing the special condition for girls does not change the number of possibilities: we have the "sex change" equality $f(n,k,l)=f(n-l,k+l,0)$
Finally it is well known that $f(n,k,0)=\binom{n+k-1}{k-1}$: a $k-1$-element subset $x_1<\dots
So: $f(n,k,l)=\binom{n+k-1}{k+l-1}$