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

321 using a single copy of each number.

#### Explanation:

A whole number is a number that sits within the range $0 - \infty$.

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:

$1 + 20 + 60 + 120 + 120 = 321$