在$a$个东西和$b$个东西中一共取出$c$个东西

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George Chrystal - Algebra_ An elementary textbook, Vol 2 Page 32 Exercise Ⅲ. 5
A selection of $c$ things is to be made partly from a group of $a$, the rest from a group of $b$. Prove that the number of ways in which such a set can be made will never be greater than when the number of things taken from the group of $a$ is next less than $(a + 1)(c +1)\over (a + b+ 2)$.
在$a$个东西和$b$个东西中一共取出$c$个东西, 方法数最大, 当从$a$中取的东西个数为$\left\lceil(a + 1)(c +1)\over (a + b+ 2)\right\rceil$时.
等价于: $f(x)=\binom{a}{x} \binom{b}{c-x}(a>x,b>c-x)$当$x=\left\lceil(a + 1)(c +1)\over (a + b+ 2)\right\rceil$时最大.
由$f(x)>f(x-1)$得$(1 + a - x) (1 + c - x)>x (b - c + x)$
消去$x^2$得$x (a+b+2)<(a+1)(c+1)$
即$x<{(a + 1)(c +1)\over (a + b+ 2)}$
所以当$x=\left\lceil(a + 1)(c +1)\over (a + b+ 2)\right\rceil$时最大.