Problem 9

Six letters are sent to six houses. If these letters are randomly delivered (each house receives one and only one letter), what is the probability that only two of them are delivered to the correct addresses?

[Problem submitted by Steve Lee, LACC Professor of Mathematics.]

Solution for Problem 9:

There are =15 ways to choose 2 letters from the 6 letters and deliver them to their correct addresses, and there are 9 ways to deliver the rest to wrong addresses (see Note*). Therefore the total number of ways to deliver only 2 letters to their correct addresses is159. There are 6! ways to deliver 6 letters to 6 houses. P==

Note*: Assume 4 letters A, B, C and D are supposed to be delivered to the 1^st, 2^nd, 3^rd and 4^th house respectively. Then there are 3 ways to deliver a wrong letter (namely B, C, or D) to the 1^st house. Say, if C is delivered to the 1^st house, then there are 3 ways to deliver a wrong letter (A, B, or D) to the 3^rd house. After this, there is only 1 way for the other 2 letters go to the wrong houses. The total number of ways that every house gets a wrong letter is the product of the number of ways each house getting a wrong letter. Therefore it is 33=9.