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There are 11 red balls & 11 blue balls in a jar. In how many ways can we pick 11 balls (at a time) from the jar when the maximum of 7 balls of one color can be chosen. For example combination can be 6 red, 5 blue or 6 blue, 5 red or 7 red, 4 blue or 4 red, 7 blue.

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Assuming all the red balls are identical & all the blue balls are identical. The number of ways is the coefficient of $x^{11}$ in \begin{eqnarray*} \color{red}{(1+x+\cdots+x^{6}+x^{7})}\color{blue}{(1+x+\cdots+x^{6}+x^{7})} \end{eqnarray*} which is $4$.

If the balls are not identical then the number of ways is the coefficient of $x^{11}$ in \begin{eqnarray*} \color{red}{(1+11x+\cdots+\binom{11}{6}x^{6}+\binom{11}{7}x^{7})}\color{blue}{(1+11x+\cdots+\binom{11}{6}x^{6}+\binom{11}{7}x^{7})} \end{eqnarray*} which is \begin{eqnarray*} 2\left(\binom{11}{6}^2+\binom{11}{7}^2 \right)=... \end{eqnarray*}

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  • $\begingroup$ You mean the 4 ways the OP listed in the question? I think we need to assume the balls are not indistinguishable. $\endgroup$ – Nuclear Wang Apr 10 at 20:13

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