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    Discrete Math and Integral Solutions

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    Solution Preview

    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 ...

    Solution Summary

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