Question

Find the numbers of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5, if no digit is repreated? How many to these will be even?

Answer 1

We can take the digits one at a time, starting at either end. Let's start from the right.
d c b a = the digits to be chosen.
For a we have 5 choices (1, 2, 3, 4, 5)
For b we only have 4 (having used one for a, and repeats not allowed)
For c we have 3.
For d we have 2.
5 * 4 * 3 * 2 = 120 choices overall
If we want the number to be even, we don't have 5 choices for a, we, are limited to the set {2, 4} there are only two digits available.
But for the remaining digits the calculation is the same.
$\frac{2}{5}$ of the numbers are even = $\frac{2}{5}\times 120=48=2\times 4\times 3\times 2$