Suppose that you have a large supply of $3$ and $7$ cent stamps. Write a recurrence relation and initial conditions for the number $S_n$ of different ways in which n cents worth of stamps can be attached to an envelope if the order in which the stamps are attached does matter.
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The ordered case, as previously noted, satisfies the recurrence $$S_{\mathrm{ord}}(n)=S_{\mathrm{ord}}(n-3)+S_{\mathrm{ord}}(n-7).$$ For the unordered case, I give the following hint: Hint (since this is flagged as homework): We can adjust this recurrence to account for the "order is not important" by adding in a correction term to account for overcounting. So, we have $$S(n)=S(n-3)+S(n-7)-[???].$$ There's quite an elegant reason for this, and I don't want to spoil this for you. The initial conditions are merely bookkeeping (they can be counted by hand, once you know the depth of the recurrence). |
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If you want to put $n$ cents worth of stamps, you can either put one $7$ cent stamp and $n-7$ cents worth of stamps (i.e. $S_{n-7}$), provided that $n \geq 7$, or one $3$ cent stamp and $n-3$ cents worth of stamps (i.e. $S_{n-3}$), provided that $n \geq 3$. As far as initial conditions are concerned, you just have to analyse what is going on for $n < 7$, which should be fairly easy. |
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