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Mathematics>Standard Multiple Choice

Read the following SAT test question and then click on a button to select your answer.

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

答案和解析

答案：D

解析：

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*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 60/6 = 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 6/10, or 3/5.