2 Variable Recursion

This problem is written as an auxiliary lemma for the solution to this problem.

Suppose $f(m,n)$ is a function that is defined for non-negative integers $m,n$ where:

$f(m,n) = 0$ if $m < n$

$f(m,0) = 1 \forall m$

$\displaystyle f(m,n) = \sum_{k=1}^{n} f(k-1,k-1) f(m-k,n-k) + f(m-1,n) \forall m \geq n$

Prove that $f(m,n) = \frac{m-n+1}{m+1} \binom{m+n}{m}$

Boys and Girls around the table

$(n+1)$ boys and $n$ girls are seated around a circular table playing a game. One boy started to put a stone on the table, and they subsequently went around the table clockwise. At each child's turn, if that child is a boy, he puts a stone on the table, but if that child is a girl, she takes away one stone from the table. They're finished when everyone has had one turn. If the table ever becomes empty (except before the first boy put his stone), they all lose the game.

Prove that there is a boy such that if he starts, they can complete the round without losing.

Furthermore, prove that there is only one such boy. That is, if any other boy starts, they would lose the game.

Harder version: suppose you now have $m+n$ boys and $n$ girls, with $m$ positive. Prove that there are exactly $m$ boys who could start a non-losing game.

Show and Change

Credit for this problem goes to a friend of Boon Leong Ng.

A ticket to a show costs 10 dollars. There are $(m+n)$ (where $m > n$) people waiting on a line. $m$ of them pays with 10 dollar bills and $n$ of them pays with 20 dollar bills. Assuming that the cashier starts out with no money, what is the probability that the cashier never runs out of change? Assuming the probability is uniform among all possible ordering of people.