How many 8-character passwords can be created from 26 lowercase letters and 10 digits, assuming that in every password there must be as many letters and numbers as?

1 Answer
Jun 14, 2018

I'll give it a go: #26^4 * 10^4 * 8!#


Reasoning that "as many letters and numbers as" means we need 4 letters and 4 numbers.

So you'll make 4 picks from each category (letters and numbers).

You can duplicate them (i.e., pick each character more than once.), for your 4 picks from the letters, you'll have

#26 * 26 * 26 * 26 = 26^4# possible selections.

For each of these, you'll have 4 picks from the digits. Similar reasoning, this works out to

#10^4# combinations of digits.

This means you'll have #26^4 * 10^4# possible combinations of 4 letters and 4 digits, and, for each of these, you now need to work out how many unique ways to arrange them.

You'll have 8 possible choices for the 1st character, 7 for the second, 6 for the third, etc. This works out to #8!# different orderings for each of your #26^4 * 10^4# combinations of letters and numbers. Multiply everything out for the answer. It's a big number.

Best I can do, but I'm far from the ultimate authority for this kind of analysis, so, as always: