Question

How many numbers more than 1 000 000 can be formed from the digits 0, 1, 2, 3, 3, 3, 4?

Answer 1

720

Explanation:

I'm going to assume we can only use each digit once.

Notice that our list of numbers is 7 digits and that 1,000,000 is also 7 digits. Since all but one of our digits is at least 1, we can place any digit in the first space, except for the 0, and have the resulting number be greater than 1,000,000.

This means that we can place any of the 6 non-0 digits in the first place:

`6xx...`

and then in the following places there are `6!` ways to select the numbers (there are 6 digits we can put in the second place, 5 in the third, etc). This gives:

`6xx6! =6xx720=4320`

And we're almost done - there are three 3s in the number sequence and they will cause duplicates that we need to divide out. We divide by the number of ways that they can internally arrange, which is `3! = 6`, so we have:

`4320/6=720`