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Derive a formula for this problem:

What are the possible number of cases to express $100$ as a sum of other positive numbers?

For example, $4$ can be expressed as a sum in the ways:

  • $1+1+1+1$
  • $2+1+1$
  • $2+2$
  • $3+1$ and lastly
  • $4$ itself

so there are $5$ cases.

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    Check [this](https://en.wikipedia.org/wiki/Partition_(number_theory)) out. – 2017-01-03
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    But can u give me a formula for it – 2017-01-03
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    Euler's recurrence formula, described in [this answer](http://math.stackexchange.com/questions/1192906/recurrence-relation-recursive-formula-closed-formula-for-partition-numbers/1718559#1718559), is a reasonably efficient way of computing the partition numbers. – 2017-01-03

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You are looking for finding integer partitions. Check this out.

There is an exact formula available as an infinite series, given in the above reference. Evaluating it, however, is highly complicated. A good approximation for integer partitions of a number $n$ is $\dfrac{1}{4n\sqrt{3}}e^\left(\pi \sqrt{\dfrac{2n}{3}}\right)$