Question

A whole number consists of the 2 4, 6, 8, and 0. How many whole numbers are possible?

Answer 1

321 using a single copy of each number.

Explanation:

A whole number is a number that sits within the range `0 - oo`.

We can make whole numbers using one, some, or all of the numbers listed. I'm going to assume that we can only use a single copy of each number, so the number `24` is ok but numbers `22` and `44` are not. So let's see what we can make.

Before we start, I'm noticing that if we make a 2-digit number starting with `0` - we end up having a single digit number. The same thing is true for a 3-digit number starting with `0` - we end up with a 2-digit number. So I'm going to skip single digit numbers and instead use 2-digit numbers starting with `0` to be those single digit numbers. In fact, the only number that we need to count at the single digit level is the number `0`, so let's count that now:

1-digit = 1

For 2-digit numbers, we have a permutation of five numbers, select 2:

`P_(5,2)=(5!)/((5-2)!)=(5!)/(3!)=(5xx4xx3!)/(3!)=20`

For 3-digit numbers, we do the same for five numbers, pick three:

`P_(5,3)=(5!)/((5-3)!)=(5!)/(2!)=(5xx4xx3xx2!)/(2!)=60`

For 4-digit numbers, we do the same for five numbers, pick four:

`P_(5,4)=(5!)/((5-4)!)=(5!)/(1!)=(5xx4xx3xx2xx1!)/(1!)=120`

And now we do the 5-digit numbers:

`P_(5,5)=(5!)/((5-5)!)=(5!)/(0!)=(5xx4xx3xx2xx1!)/(1)=120`

And now we add them up:

`1+20+60+120+120=321`