PRMO 2012 question 6 postman letters

Q6: A postman has to deliver five letters to five different houses. Mischievously, he posts one letter through each door without looking to see if it is the correct address. In how many different ways could he do this so that exactly two of the five houses receive the correct letters?

Solution:
Let's do it via an example.
Let's say 5 houses are A,B,C,D,E.
Let's pick 2 correct houses A,B and compute how many ways are there to deliver remaining 3 letters incorrectly. Whatever the answer, we will multiply it with 5C2(=10) since there are 5C2 ways to choose the 2 correct houses.

So now distribute 3 letters incorrectly to C,D,E.
To the first house C there are 2 ways to give the wrong letter(D or E). Let's pick D.
Now to remaining 2 houses D,E there is only 1 way to give the incorrect letter. So in total there are 2x1x1 ways.
D,E,C
E,C,D

So answer is 10x2 = 20.

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