Discussion
A bag contains 5 red balls, 4 black balls, and 3 white balls. Then the number of ways of selecting three balls at random that contains at least one white ball is
Answer: D
There are 136 ways.
Count the complement. There are 12 balls in all, 9 of them not white:
\[ \begin{aligned} \binom{12}{3} - \binom{9}{3} &= 220 - 84 \\ &= 136 \end{aligned} \]
Direct counting agrees: one white \(3 \times \binom{9}{2} = 108\), two white \(3 \times 9 = 27\), three white 1; total 136. 220 is the count of all selections, with no condition at all.
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