Arrangement of Counters-II
how many different ways may counters be placed on the squares of an grid so that no two counters are in the same row or in the same column?
Hint : - You may decide to solve this first.
The answer is 25200.
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1 solution
There are four counters. Let's put it piece by piece on the board. The first piece can be placed anywhere on the board; thus we have ( 5 ) ( 1 0 ) = 5 0 ways to do that. Now for the second counter, it can't be on the same row and column of the first counter, so we have ( 4 ) ( 9 ) = 3 6 ways to do that. The third counter can't be on the same row and column of the first and second counter, so we have ( 3 ) ( 8 ) = 2 4 ways to do that. The fourth counter can't be on the same row and column of first, second and third counter, so we have ( 2 ) ( 7 ) = 1 4 ways to do that.
total number of ways = 4 ! ( 5 0 ) ( 3 6 ) ( 2 4 ) ( 1 4 ) = 2 5 2 0 0
Note that we divided it by 4 ! because the order in which the counters are placed is not important.