In a speaking contest, there are six finalist.One winner is to be selected for the first prize, two winners for second prize and three winners for third prize.What is the number of ways the prizes can be awarded?

1 Answer
MathFact-orials.blogspot.com
Jul 18, 2018

60

Explanation:

Let's think of this problem this way - if we arranged the six finalists on a row of six chairs, we'd have #6! = 720# ways to arrange them.

However, the 2nd and 3rd chairs are, in essence, the same because they are 2nd place. And the 4th, 5th, and 6th chairs are also essentially the same, being 3rd place. So to account for the duplicates arising from the chairs being essentially the same, we divide by the ways each of those groups of chairs are the same (so we divide by #2!# for the 2nd place chairs and #3!# for the third place chairs). This gives:

#(6!)/(3!2!)=720/(2xx6)=60#

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