Password combinations

Question

Can be 5 or 6-character long passwords. The characters must be from lowercase letters or digits. Each password must contain at most two lowercase-letters and contains no repeated digits. How many valid passwords are there?

Answer 1

Let P(n,k) denote the set of valid passwords of length n, containing k letters.

P(n,k) are constrained by n = 5 or 6, and k = 0, 1, or 2.

Because of formatting constraints here, I will use the notation

C(n,k) means "n chose k"

That is,

C(n,k) = n!/(n!(n-k)!)

We calculate the number of passwords in the set P(n,k).

We can chose the k positions for the k letters in C(n,k) ways.

For each of those we can chose the letters for those k positions in 26^k ways.

For each of those we can chose unique digits for the remaining n-k position in

10!/(10-n+k)! ways.

Consequently

P(n,k) = C(n,k)×26^k×10!/(10-n+k)!

So you need to calculate the size of P(n,k) for the six combinations of n and k,

and then add up the six results.