Multiple choice

6 married couples are standing in a room. if 4 people are chosen at random, then the chance that exactly one married couple is among the 4 is-

  1. $\cfrac{16}{33}$
  2. $\cfrac{8}{33}$
  3. $\cfrac{17}{33}$
  4. $\cfrac{24}{33}$
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A Correct answer
Explanation

There are 12 people total (6 couples). The total ways to choose 4 people is 12C4 = 495. To choose exactly one couple, we select 1 couple out of 6 (6C1) and then 2 people from the remaining 5 couples such that they are not a couple (5C2 * 2^2). This gives 6 * 10 * 4 = 240. The probability is 240/495 = 16/33.

AI explanation

The total number of ways to choose 4 people from 12 is 12C4, which equals 495. The number of ways to choose exactly one married couple and two other people is 6C1 x 10C2. Solving 10C2 gives 45, but we must subtract the 5 remaining married pairs to ensure we pick two individuals who are not married to each other, leaving 40 valid pairs for the remaining two spots. Thus the favorable outcomes are 6 x 40 = 240, and the probability is 240/495, which simplifies to 16/33.

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