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Using pennies, nickels, dimes, and quarters. How any ways does it take you to make 69 cents?

 Jun 13, 2021
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Let 
\begin{eqnarray*}
f_1 &=& 1+x+x^2+\dots+x^{68}+x^{69} \\
f_5 &=& 1+x^5+x^{10}+\dots+x^{60}+x^{65} \\
f_{10} &=& 1+x^{10}+\dots+x^{50}+x^{60} \\
f_{25} &=& 1+x^{25}+x^{50}.
\end{eqnarray*}
Find the term for $x^{69}$ in the expansion of $f_1f_5f_{10}f_{25}$.  The answer is the coefficient of the term for $x^{69}$.
 

 Jun 13, 2021

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