Repetitions allowed and not allowed

Question

You are asked to select a password for a Web site. It must consist of five lowercase letters and two digits in any order. How many possible such passwords are there if

a) repetitions are allowed?

b) repetitions are not allowed?

For the first part of the problem one can choose $5$ letters in $26^5$ different ways, $10^2$ ways to choose $2$ digits on. In the book they say that the answer is

$$26^5\cdot 10^2\cdot{7\choose 2}$$

I suppose that the binomal states the number of ways to arrange $2$ tokens among $7$?

For the second part, the answer is

$$(26)_5\cdot (10)_2\cdot{7\choose 2}$$

What does that subnotation even mean? Can someone briefly explain the difference between using the binomal and using the multiplication principle?

Also, how does the repetition make a difference in this case?

Answer 1

We multiply with $\binom{7}{2}$ because we choose two places out of seven for the digits and then we know where to put the letters.

As for the second case, if repetition is not allowed you have $26\cdot 25\cdot 24 \cdot 23 \cdot 22$ for the letters because the number of choices decreases, and also for the numbers you have $10 \cdot 9 $ because you cannot repeat the digit you already chose.

Answer 2

For the first part, there are $26^5$ ways to choose the letters, in order, and $10^2$ ways to choose the digits, in order. There are $\binom{7}{2}$ ways to determine which positions to put the digits in.

The subscript in the second part is the falling factorial. (There are a lot of different notations for this.) Anyway, $(26)_5=26\cdot 25\cdot 24\cdot\ 23\cdot 22$