# How many ways are there to place 5 checkers on a 5x5 board

or, similarly, given 25 switches, how many ways are there to turn on 5 of them...

I'm not interested in the number, I want to know how to calculate it...

• How many ways are there of choosing 5 elements out of a set of 25? Commented Jul 19, 2015 at 20:28
• Instinctively I think ... if we have 25 numbers, and we pick at random, it's 25*24*23*22*21. But order doesn't matter (1,2,3,4,5 is the same as 1,2,3,5,4), so it's... less than that? Commented Jul 19, 2015 at 20:38
• 25*24*23*22*21, and then there are 5! identical orderings
– qwr
Commented Jul 19, 2015 at 20:39
• @qwr: Do you want to close the deal and write it down, or should I? Commented Jul 19, 2015 at 22:26
• @Gary. You can try answering, but the question looks like it'll be closed soon
– qwr
Commented Jul 20, 2015 at 2:27

You are looking for the number of combinations of five things out of twenty-five, written $25 \choose 5$ (there are other ways that are used). You have $25$ squares for the first checker, $24$ for the second, down $21$ for the fifth, but we don't care what order we put the checkers down in, so we divide by $5!$ orders we could have chosen giving ${25 \choose 5}=\frac {25 \cdot 24 \cdot 23 \cdot 22 \cdot 21}{5!}=\frac {25!}{5!20!}$