Question

How many numbers with three distinct digits have digits from {1,2,3,4,5} and a digit is 3?

Answer 1

36

Explanation:

We have the numbers `1, 2, 3, 4, 5` to choose from to make 3 digit numbers and one of those numbers needs to be a 3. That means we can look at this problem this way - of the three digits, we can put a 3 in the first digit:

`3XX`

leaving four digits to put into the `X` places. Since this will also be true with the 3 here:

`X3X`

and here

`XX3`,

we can then say that whatever result we get for the first option we can then multiply by 3 to find all the options.

As far as choosing the remaining two numbers , we can either find that directly:

`4xx3=12`

or go via the permutation route (permutation because we care about order) to get to:

`P_(n,k)=(n!)/((n-k)!); n="population", k="picks"`

`P_(4,2)=(4!)/((4-2)!)=(4!)/(2!)=(4xx3xx2!)/(2!)=4xx3`

Putting it all together, we get:

`(4xx3)xx3=36`