Question

A group of 7 finalist in a ceramic art competition are to be awarded 4 prizes..first, second and so on. How many different ways are there to award these prizes

Answer 1

Solution

The first prize can be awarded to any one of 7 finalists in 7 ways. Once the first prize is awarded to a particular finalist, that finalist is not eligible for consideration of the remaining three prizes. So, the second prize can be awarded to only any one of 6 remaining finalists and this can be done in 6 ways. Once the first prize is awarded to a particular finalist, and the second prize is awarded to a different finalist, these two finalists are not eligible for consideration of the remaining two prizes. So, the third prize can be awarded to only any one of 5 remaining finalists and this can be done in 5 ways. Once the first prize is awarded to a particular finalist, the second prize is awarded to a different finalist, and the third prize is awarded to yet another different finalist, these three finalists are not eligible for consideration of the remaining fourth prize. So, the fourth prize can be awarded to only any one of 4 remaining finalists and this can be done in 4 ways.

Now, for every one way of awarding the first prize, there are 6 ways of awarding the second prize. Similarly, for every one way of awarding the first prize and the second prize, there are 5 ways of awarding the third prize, and for every one way of awarding the first prize, the second prize and the third prize, there are 4 ways of awarding the fourth prize.

Thus, in total there are (7 x 6 x 5 x 4) = 840 ways of awarding these 4 prizes. ANSWER