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If there are 7 runners in a race, how many different orderings of 1st, 2nd, and 3rd place can there be?

I did 7*6*5 to get 210. I did this because there are 7 runners that can get first, after one finishes there are 6 that are able to get second, and 5 to get third. Am I correct?

Thanks.

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1 Answer

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Just for the reason that, you do not have any further doubt on this.

If you have studied Permutations & Combinations or Combinatorics, you will know, actually, the 1st, 2nd and 3rd places can be selected in $^7P_3$ or ${7\choose 3}\times 3!$. Here, what we are doing is, we are finding the number of ways of choosing 3 elements from the 7 given elements to be our 1st, 2nd and 3rd place holders and since the order matters here, i.e., how the 3 chosen elements are arranged to be 1st, 2nd and 3rd matters here, we multiply it by $3!$.

I hope you find something of your necessity here.

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