Question

How many 5 digits numbers can we get from numbers 1,3,5,18,70?

Answer 1

66

Explanation:

We're given 3 single-digit numbers and 2 two-digit numbers and asked to make unique 5 digit numbers.

To try to keep things clear, I'm going to label the digit places by letters, and so a is the first digit, b the second, and so on to e being the last digit.

There are two basic scenarios we need to examine - one where we use all the single digit numbers and 1 of the two-digit numbers, and the other where we use both the two-digit numbers and use 1 of the single digit numbers.

3 single-digit, 1 double-digit

Let's first look at how we can place the double digit number - it can sit in digit place ab, bc, cd, de - 4 places.

There are 2 of the double-digit numbers.

The 3 single-digits numbers can be placed any where around the double number, so that's `3! = 6`

And so in total, there are

`4xx2xx6=48` ways to place the digits in this fashion.

2 double-digit, 1 single digit

The placement for the double-digit numbers is fairly restrictive. We can put one of them in ab and have the other in cd or de. We can also place one of them in bc and have the other in de. That's 3 unique ways to place them.

There are 2 double-digit numbers to choose from, so their arrangement within these 3 choices is `2! = 2`.

And lastly there is the single-digit number. There will only be one place for it to sit, so that's `1! = 1`.

There are 3 to pick from.

And so in total, there are

`3xx2xx1xx3=18` different ways to arrange the digits in this fashion.

Total

In total, there are

`48+18=66` different ways to arrange the digits.