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Problem 76

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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?

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加和计数

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？

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