# 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?

Jun 14, 2018

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

#### Explanation:

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.)

...so, for your 4 picks from the letters, you'll have

$26 \cdot 26 \cdot 26 \cdot 26 = {26}^{4}$ 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 ${26}^{4} \cdot {10}^{4}$ 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 ${26}^{4} \cdot {10}^{4}$ 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:

GOOD LUCK