From (6) red and (5) blue balls, in how many ways can (2) red and (1) blue ball be selected?

Correct answer: (75)

The two colors are separate groups, so first choose two red balls from six and then choose one blue ball from five. The red selection can be made in \(\binom62=\frac{6\times5}{2}=15\) ways. The blue selection can be made in \(\binom51=5\) ways. Since every red choice can be combined with every blue choice, multiply them: \(15\times5=75\).

The balls of the same color are selected without regard to order, so combinations are appropriate. We do not arrange the selected balls, because the question asks only which balls are chosen. Therefore the total number of selections is 75, which is option D. A result such as 30 would undercount the possible pairs of red balls.