Question

For The Questions, a club has 10 members.

a. Find the number of different slates of 3 officers (list of president, vice-president, and treasurer) that the club could have for officers.

b.Find the number of different slates of 4 officers (list of president, vice-president, secretary, and treasurer) that the club could have for officers.

Answer 1

Are you studying combinations and permutations?

Answer 2

Yes

Answer 3

Is this question a combination or a permutation? What do you think?

Answer 4

Remember that order doesn't matter in a combination but order does matter in a permutation.

Answer 5

i think the first one is a permutation

Answer 6

Right you are. In fact, they both are. Can you see that?

Answer 7

Yes, because they have to be in a certain order, right?

Answer 8

So, in the first question, you need to calculate\[_3 P_{10}\]Can you do that?

Answer 9

I'm not sure how too

Answer 10

Actually, my notation was backwards. Sorry. The general equation for permutations is\[_{n}P_r=\frac{ n! }{\left( n-r \right) !}\]And in the first question n=10 and r=3. Can you take it from here?

Answer 11

I got -2 that's probably not correct is it?

Answer 12

Not quite. The equation becomes\[_{10}P_3=\frac{ 10! }{ 7! }\]If you write out the factorial multiplications you'll see that all the factors from 7 down will cancel.

Answer 13

Something like\[_{10}P_3=\frac{ 10\times9\times8\times7\times6\times5\times4\times3\times2\times1 }{ 7\times6\times5\times4\times3\times2\times1}\]

Answer 14

See. Everything from 7 down will cancel out leaving\[_{10}P_3=10\times9\times8=?\]

Answer 15

720? that's what I got from the multiplication

Answer 16

Excellent. Now try the second question by yourself. You need to calculate\[_{10}P_4\]

Answer 17

5040?

Answer 18

Terrific. Well done.

Answer 19

Thanks!

Answer 20

You're welcome