Gambling Game

문제

The Ionian Commission on Procuring Cash has come up with a new gambling game to raise funds for the government. The game is played as follows: Each week, the government will televise a set of $mm$ balls (numbered $1\dots m1\; \backslash ldots\; m$) being selected one at a time without replacement. Anyone who wants to play will have to buy a game card. Each card contains $nn$ squares (where $n\le m\mathrm{/}2n\; \backslash leq\; m/2$) and in each square are two numbers between 1 and $mm$. No number appears more than once on a card. A sample card is shown in Figure E.1.

Figure E.1: Sample game card with m = 10, n = 4 and p = 5.

After each ball is selected, players cover any square which contains that number (there will be at most one such square on any card). Each game card also has a number $pp$ printed on it, and a contestant wins if he or she covers all $nn$ squares after exactly $pp$ ball selections (i.e., prior to the $p^{\text{th}}p^\{\backslash text\{th\}\}$ selection, they only had $n-1n-1$ squares covered). Before issuing cards to its citizens, the government would like to get an idea of the likelihood of winning for various values of $m,nm,\; n$ and $pp$ so they can set up the payoffs appropriately. They have procured you to write a program to solve this problem.

입력

Input consists of a single line containing 3 integers $m,nm,\; n$ and $pp$, as described above, where $2\le m\le 332\; \backslash leq\; m\; \backslash leq\; 33$, $0\le n\le m\mathrm{/}20\; \backslash leq\; n\; \backslash leq\; m/2$ and $0\le p\le m0\; \backslash leq\; p\; \backslash leq\; m$.

출력

Output the probability of winning on the $p^{\text{th}}p^\{\backslash text\{th\}\}$ selection as a fraction x/y in simplest form.