smc

Suppose that $t$ tenors and $b$ basses are selected. The requirements are $t\equiv b(\mathrm{mod}4)$ and $(t,b)\ne (0,0).$ It follows that $b^{\prime }=8-b$ basses are not selected. Since the ordered pairs $(t,b)$ and the ordered pairs $(t,b^{\prime })$ have one-to-one correspondence, we consider the ordered pairs $(t,b^{\prime })$ instead. The requirements become $t\equiv 8-b^{\prime }(\mathrm{mod}4)$ and $(t,8-b^{\prime })\ne (0,0),$ which simplify to $t+b^{\prime }\equiv 0(\mathrm{mod}4)$ and $(t,b^{\prime })\ne (0,8),$ respectively. As $t+b^{\prime }\in \{0,4,8,12\},$ the total number of such groups is $$\begin{array}{rl}N& =\left(\genfrac{}{}{0ex}{}{14}{0}\right)+\left(\genfrac{}{}{0ex}{}{14}{4}\right)+\left[\left(\genfrac{}{}{0ex}{}{14}{8}\right)-1\right]+\left(\genfrac{}{}{0ex}{}{14}{12}\right)\\ & =\left(\genfrac{}{}{0ex}{}{14}{0}\right)+\left(\genfrac{}{}{0ex}{}{14}{4}\right)+\left[\left(\genfrac{}{}{0ex}{}{14}{6}\right)-1\right]+\left(\genfrac{}{}{0ex}{}{14}{2}\right)\\ & =1+1001+[3003-1]+91\\ & =4095,\end{array}$$ from which $N\equiv \boxed{95}(\mathrm{mod}100).$