From (6) red and (7) blue balls (5) balls are to be selected with exactly (2) red balls. How many ways are there?

Correct answer: 525

Out of the 5 balls, exactly 2 must be red, so the remaining 3 must be blue. The number of ways to choose 2 of the 6 red balls is \(\binom{6}{2}=15\), and the number of ways to choose 3 of the 7 blue balls is \(\binom{7}{3}=35\). Hence, the total number of selections is \(\binom{6}{2}\binom{7}{3}=15\times35=525\). A value such as 315 does not correctly account for the required numbers of red and blue balls. Exam tip: when “exactly” is given, first fix the number required from each group and then multiply the independent selections.