Question

From a pool of 12 candidates, the offices of president, vice president, secretary, and treasurer will be filled. In how many different ways can the offices be filled?

Answer 1

11,880

Explanation:

We can find the number of ways in a couple of different ways:

• work out the number by examining each seat
• use the Permutation formula

Examine each seat

Let's start with the President's seat. How many people can fill that? 12

And then the Vice President's seat - how many can fill that? 1 person is already in the President's seat, so there's 11 people who can.

And then then the Secretary's seat - 2 people are already in seats and so 10 could fill this one.

And lastly the Treasurer's seat - 3 people are already in seats and so 9 could possibly sit here.

So that's `12xx11xx10xx9=11,880`

Permutation

It'd be great if we could express the above multiplication in terms of factorials, because factorials are set figures and so we don't need to work out the multiplication each time. Is there a way to do so?

Notice that in our multiplication above, we have `12xx11xx10xx9`. What if we could have the full factorial? To have that, we'd need to have the multiplication of consecutive natural numbers down to 1 (and we'll divide by that as well to maintain the value):

`(12xx11xx10xx9xx8xx7xx6xx5xx4xx3xx2xx1)/(8xx7xx6xx5xx4xx3xx2xx1)`

And we can express this as:

`(12!)/(8!)=(479,001,600)/(40,320)=11,880`

https://www.mathsisfun.com/numbers/factorial.html

Notice that `12-8=4`, which is the number we sat in the officer's chairs, so we can say:

`(12!)/((12-4)!)` - and this is the equation we go through with a permutation:

`P_(n,k)=(n!)/((n-k)!); n="population", k="picks"`