Question

1. How many 4-digit codes can be made if no digit can repeat? 2. How many ways can 5 bikes be displayed in a store window from a group of 12?

Answer 1

1. `=5040`, 2. `=792`

Explanation:

First problem: 4-digit codes where no digit repeats:

The first digit can be anything from 0-9, so we have 10 choices.
The second digit can by anything except the first digit, so we have 9 choices
The third digit we have 8 choices
The fourth digit we have 7 choices:

`10xx9xx8xx7=5040`

Second problem: There are 12 bikes to choose from and we want to display 5. Since choosing bikes 1, 2, 3, 4, 5 is the same as choosing bikes 5, 4, 3, 2, 1 we have a combinations problem.

`C_(12,5)=(12!)/(5!(12-5)!)=(12!)/(5!7!)`

and now let's evaluate it:

`(12!)/(5!7!)=(12xx11xx10xx9xx8xx7!)/(5xx4xx3xx2xx7!)`

`(cancel12xx11xxcancel10xx9xx8xxcancel(7!))/(cancel5xxcancel4xxcancel3xxcancel2xxcancel(7!))=11xx9xx8=792`