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I am looking for a formula or at least something to use when trying to compute how many ways I can make change for an amount. Example: there are $3$ ways to give change for $4$ if you have coins with denomination $1$ and $2$: $1+1+1+1, 1+1+2, 2+2$. What formula should I use? |
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This problem is called integer Partitioning. There is a recursive formula to list all possible partitions of an integer. Given a number of 4 you could generate the following sets of values: 4 3 1 2 2 2 1 1 1 1 1 1 Please refer to Wiki-Partitions for details. Unfortunately, it is not a trivial calculation. There is another problem related to this that is called Money Changing Problem or Change Making Problem , where you'd be interested in specific values rather than getting all possible values. For example changing Five Dollars in a country where there is no 4 dollar notes or coins. For such problems you may use R, Mathmatica, etc. |
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The formula to use is to find the coefficient of $X^a$ (where $a$ is your amount) in the expansion as a power series in $X$ of $$ \frac1{(1-X^{d_1})(1-X^{d_2})\cdots(1-X^{d_n})} $$ where $d_1,d_2,\ldots,d_n$ are the denominations of your coins. In principle this is just an algebraic reformulation that does not help you at all: finding the coefficient in the straightforward way (writing each $\frac1{1-X^{d_i}}$ as a geometric series and multiplying out) merely gives you the combinatorial problem you started with. However, as I explained in this answer to a related question, you can actually perform these multiplications very easily without even expanding each $\frac1{1-X^{d_i}}$ in the first place, by a simple loop updating a sequence of coefficients of your current power series (truncated after the term that interests you). |
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