Problem 76

Counting Summations

It is possible to write five as a sum in exactly six different ways:

$\begin{aligned} 4 + 1 \\ 3 + 2 \\ 3 + 1 + 1 \\ 2 + 2 + 1 \\ 2 + 1 + 1 + 1 \\ 1 + 1 + 1 + 1 + 1 \end{aligned}$

How many different ways can one hundred be written as a sum of at least two positive integers?

将 $5$ 写成至少两个正整数的和，有 $6$ 种不同的方式：

$\begin{aligned} 4 + 1 \\ 3 + 2 \\ 3 + 1 + 1 \\ 2 + 2 + 1 \\ 2 + 1 + 1 + 1 \\ 1 + 1 + 1 + 1 + 1 \end{aligned}$

将 $100$ 写成至少两个正整数的和，有多少种不同的方式？