The Official SAT Question of the Day™

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Of 5 employees, 3 are to be assigned an office and 2 are to be assigned a cubicle. If 3 of the employees are men and 2 are women, and if those assigned an office are to be chosen at random, what is the probability that the offices will be assigned to 2 of the men and 1 of the women?

A 1/3
B 2/5
C 1/2
D 3/5
E 2/3

Correct Answer: D

Here's Why:

Of the 5 office workers, 3 are to be assigned an office. This is an example of combinations: to find the number of ways of choosing 3 of the 5 workers, you can count the number of ways of selecting the workers one at a time and then divide by the number of times each group of 3 workers will be repeated.

There are 5 ways of choosing the first worker to get an office. Then there will be 4 ways of choosing the second worker to get an office, and 3 ways of choosing the third worker. This is a total of 5 × 4 × 3 = 60 possibilities. In these 60 possible selections, each distinct group of 3 workers will occur 3 × 2 × 1 = 6 times. (There are 3 possibilities for the first worker chosen from the group, 2 for the second worker chosen, and only 1 for the third.) Therefore, there are = 10 different ways the 3 workers who get an office can be chosen from the 5 workers.

How many of these 10 possible groups of 3 workers consist of 2 men and 1 woman? From the 3 male workers, 2 can be chosen in 3 different ways. There are 2 possibilities for the female worker. Therefore, 3 × 2 = 6 of the groups of 3 workers consist of 2 men and 1 woman.

Since there are 10 different ways the 3 workers who get an office can be chosen, and 6 of these possible groups of 3 workers consist of 2 men and 1 woman, the probability that the offices will be assigned to 2 men and 1 woman is , or .


Difficulty: Hard
Question Type: Standard Multiple Choice

(Mathematics)