Question

How many ways can a teacher arrange four students in the front row with a total of 30 students?

Answer 1

I'm confused

Answer 2

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

Answer 3

Idk

Answer 4

Permutation?

Answer 5

Does the order in which the four students placed in the front row matter?

Answer 6

Yes?

Answer 7

So let's say that Bill, Jim, Sue, and Betty are chosen to sit in the front row. Does it matter in what order they sit?

Answer 8

No

Answer 9

Great. So, as long as those four are chosen, it doesn't matter in which order they're chosen. If the order doesn't matter, is that a combination or a permutation?

Answer 10

Permutation?

Answer 11

Sorry, no. If the order doesn't matter, then it's a combination. If the order does matter, then it's a permutation. So this question is about calculating a combination. Do you know how to do that?

Answer 12

No

Answer 13

The number of combinations of choosing r items out of a set on n items is given by\[_{n}C _{r}=\frac{ n! }{ r!\left( n-r \right)! }\]Do you understand this equation?

Answer 14

No

Answer 15

OK. In this question, n is the total number of people. How many is that?

Answer 16

30

Answer 17

Yup. And r is the number of people that are chosen to sit in the front row. How many is that?

Answer 18

4

Answer 19

Right. So the calculation becomes\[_{30}C _{4}=\frac{ 30! }{ 4!\left( 30-4 \right)! }=\frac{ 30! }{ 4!26! }\] Can you calculate this answer?

Answer 20

Not really

Answer 21

Do you know what the factorial symbol (!) means?

Answer 22

4!x26! ?

Answer 23

Yes

Answer 24

I'm using a scientific calculator

Answer 25

Great. So you have\[_{30}C _{4}=\frac{ 30\times29\times28\times27\times26\times25\times24\times...\times2\times1 }{\left( 4\times3\times2\times1 \right)\left( 26\times25\times24\times...\times2\times1 \right) }\]Make sense?

Answer 26

If so, there are a lot of common factors in the numerator and denominator that cancel out.

Answer 27

Still there?

Answer 28

Yes sorry had to do something

Answer 29

Cancel out the common factors and calculate the result. What do you get?