MHB How many possible 7-place license plates are there with 3 letters and 4 digits?

[iamjokerface]

Hello,

I am stuck on this one problem. The problem asks:

How many different 7-place license plates are possible when 3 of the entries are letters and 4 are digits? Assume that repetition of letters and numbers is allowed and that there is no restriction on where the letters or numbers can be placed.

The way I approached the problem was:

There are 7C3 ways of choosing the places for letters, and in each letter place, there can be 26 choices. So 7C3*26*26*26 for the letters.
Then there are 7C4 ways of choosing the places for digits, and in each digit place, there are 10 choices, so 7C4*10*10*10*10.
To get the total possibilities, I multiplied the two.

I checked the answer but I am not getting the correct answer.

Where have I gone wrong?

 

[Ackbach]

I think the problem is that letters are indistinguishable from one another, as are numbers. I would consider it more like this:

There 7 places to choose the first letter, 6 for the second, and 5 for the third. Once you've chosen the letter locations, the rest are all numbers. So I would probably have an expression like: $7\cdot 6\cdot 5\cdot 26^3 \cdot 10^4$. Does that work?