First determine how many ways 6 people can be distributed among the 12 months.

Any person could be born in any of the 12 months, so there are 12^6 ways these 6 births could be distributed among the 12 months.

Now on to selecting the two months. There are 12!/(2!10!) ways of selecting two months from the 12, or 66.

Now distribute the 6 people between the two months. In a manner similar to distributing among 12 months, each person has a choice between the 2 months, so the total is 2^6 or 64.

Reading the question carefully indicates that the 6 people have to be distributed to EXACTLY 2 months. However, the 64 total above also includes 2 scenarios where all 6 people are born in one of the two months but not the other. So the 64 has to reduced to 62.

So there are 66 x 62 ways = 6x11x2x31 = 12x11x31 = 12 x 341 ways this could happen.

The probability of this happening is (12x341)/12^6 = [spoiler] D, 341/12^5 [/spoiler]

BTGmoderatorDC wrote:Probability that the birthdays of 6 different persons will fall in exactly 2 calendar months is

A. 341/12^6
B. 352/12^6
C. 352/12^5
D. 341/12^5
E. 371/12^6

Recall that probability = (# of favorable outcomes)/(total # of outcomes). Since there are 6 people and they can be born in any of the 12 months, so the total number of outcomes is 12^6, which will be the denominator of our probability.

As for the numerator of our probability, the number of favorable outcomes, first let's determine the number of ways one can select two (different) months out of 12: 12C2 = (12 x 11)/2 = 66. Now for each pair of months selected, since each of the 6 people can be born in either one of these two months, there are 2^6 = 64 possible combinations (for example, let say the two months are May and June, we could have 5 people born in May and 1 person born in June, or 2 people born in May and 4 people born in June, etc). However, these 64 possible combinations include the two combinations in which all 6 people are born in one month but not the other (that is, all 6 people can be born in May but not in June and vice versa). Therefore, we need to deduct 2 from 64. So the number of favorable outcomes is:

66 x (64 - 2) = 66 x 62

Therefore, the probability that the birthdays of 6 different persons will fall in exactly 2 calendar months is:

(66 x 62)/12^6 = (6 x 11 x 2 x 31)/(12 x 12^5) = (11 x 31)/12^5 = 341/12^5

Answer: D