No. of ways to pick 11 balls from a jar of 22 balls

Question

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.

Answer 1

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*}