Rough figure

Six P's have to be placed in the squares of the diagram given above such that each column contains at least one P.In how many ways can this be done?


The answer is 26.

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2 solutions

Pulkit Gupta
Dec 20, 2015

Whenever there's an atleast in the wording of the question, we can use case cancellation method.

Total number of ways of filling 6 Ps in 8 squares is equal to 8 ! 2 ! 6 ! \Large \frac{8!}{2! 6!} = 28.

Now notice that if we have to fill 6 Ps in only two columns, there are 2 ways to do that ( the middle column+one of the two side columns are taken).

To fill them in one column would be an impossible case.

Now cancelling 2 from 28, we get 26 as the answer.

Aiman Rafeed
Dec 29, 2015

There are 8C6 ways to put the P's. The only "2 cases" when there will be no P in at least one column is either the rightmost column being empty or the leftmost column being emty. Hence the answer is 8C6-2=26.

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