A club with 15 members is to choose four officers–president, vice...

Question

A club with 15 members is to choose four officers–president, vice president, secretary, and treasurer. In how many ways can these offices be filled?

Accepted Answer

Welcome back. I'm so glad you're here. We've got 12 budding artists entered into this drawing competition and we want to know how many different ways. 1st, 2nd and 3rd prize can be awarded. So the first question we ask is does order matter? Does it matter who comes in first, Who comes in second and who comes in third? It does. So we know we're dealing with permutation, permutation notation is written, we have end things taken R. At a time And things. How many things do we have? We've got 12. The things are the students the 12. And how many are we taking at a time? We're grabbing from that 12, 3 at a time to get 1st, 2nd and 3rd prize. So we can rewrite this. We've got 12 things taken three at a time. We recall from previous lessons that the permutation formula is N factorial in the numerator and in the denominator parentheses n minus R. And parentheses factorial. So now we can just fill that in. We've got 12 factorial in the numerator over parentheses N 12 minus R three and parentheses factorial. We can simplify that numerator is 12 factorial, 12 minus three is nine. So nine factorial in the denominator. Now, I'm going to write out in long foreign form the steps we're going to be taking so that in the future you could skip this step. So 12 factorial. That means we've got 12 times 11 times 10 times nine times eight times seven times six times five times four times three times two times one, that's the numerator. And in the denominator we've got nine times eight times seven times six times five times four times three times two times one. Great. Starting in the denominator, starting with that nine we've got a nine in both the numerator and the denominator. Those cancel out and same with the eights and sevens and so on. These factors will all cancel out from nine down to one And we can think of that in a different way so that we don't have to write all that out. We know that the numerator could be written as 12 times 11 times times nine factorial over nine factorial. And as we just demonstrated above those nine factorial cancel out. So both ways we've got 12 times 11 times We can put that into a calculator and that gives us an answer of 1320 that matches with answer choice D. While done, we'll catch you on the next one.

A club with 15 members is to choose four officers–president, vice president, secretary, and treasurer. In how many ways can these offices be filled?

Key Concepts

Permutations

Factorial

Combinatorial Counting

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