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Discrete Math and Integral Solutions

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Solution Summary

We solve the problem by mapping the problem to one involving identical particles that can be in different states.

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It is convenient to define new variables:

y_{i} = x_{i} - 2 i

Then y_{i} >= 0, and

y_{1} + y_{2} + ....+y_{k} = n - k (k+1)

So, we see that there are no solutions if k (k+1) > n. If k (k+1) <= n, then we can find the number of solutions as follows. Put m = n - k(k+1). A solution of

y_{1} + y_{2} + ....+y_{k} = m

can be interpreted as a configuration of m identical particles that can be in k different states. A state of the m- paticle system can then be specified by declaring how many particles there are in state number r for each r ranging from 1 to k. We can denote these numbers ...

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