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I can find the answer by listing all the possible ways but I want to know a more logical way to solve this problem. Can someone help me?

Air Mike
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Andrew
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    Start by choosing how many $$5$ bills there are. There has to be an odd number of $$5$ bills(by parity). Then the number of $$2$ bills will follow once you choose the number of $$5$ bills. – Aiden Chow Sep 15 '20 at 22:58
  • Would you be interested in seeing a general method to figure out recursions for this kind of thing? – paul garrett Sep 16 '20 at 01:10
  • It can help, https://math.stackexchange.com/questions/80822/solutions-to-linear-diophantine-equation-15x21y-261 – Lion Heart Sep 16 '20 at 07:02

2 Answers2

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Solving this equation $5x+2y=69$ in positive integers

$2y=69-5x$

x is an odd number and $0<x<14$, since $69-5x>0$ is a multiple of 2 and positive that is an even number

There are 7 solutions.

Lion Heart
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  • Sorry, this doesn't help the questioner much. E.g., why "7", in terms of the other parameters? What process did you carry out...? – paul garrett Sep 16 '20 at 01:05
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It is easy to find the first two dozen terms and a lookup in OEIS finds that it is sequence A008616 which has a generating function of $1/((1-x^2)(1-x^5))$. The entry also has a formula $a(n) = \lfloor n/10+ (3+(-1)^n)/4\rfloor.$ Thus $a(69)=7.$ If you want a list of these solutions you can use an extended g.f. of $1/((1-x^2)(1-x^5y))$ and get the coefficient of $x^n$ as a polynomial in $y$.

Somos
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