A password consists of seven characters Count the number of

A password consists of seven characters. Count the number of possible passwords that can be created in each of the following scenarios. You may assume numbers are digits 0 through 9 and letters are a through z (lower case only). (a) Each character must be a letter (b) The first four characters must be letters, and the last three characters must be numbers. (c) Each password contains exactly four letters and exactly three numbers (in any position). [HINT: Think of a sequence of steps that will produce such a password, and apply the multiplication rule]

Solution

a) Over here, not only the combination but also the order plays an important role.

There are 26 alphabets and 10 numbers.

So, if all the characters are alphabets and repetition is allowed, then each of the 7 spots can be filled in 26 different ways.

So total number of ways = 26 * 26 * . . . .26 (7 times) = 26⁷ = 8,031,810,176 ways

b)

Again assuming that repetitions are allowed, the first 4 spaces can be filled in 26 ways each and the last 3 spot in 10 ways each

So, total number of ways = 26⁴ * 10³ = 456,976,000 ways

c)

Each password containing 4 letters and 3 number exactly =  26⁴ * 10³ = 456,976,000 ways

however, those combinations can be rearranged amongst themselves to create more permutations

This can be done in 7! ways = 5040 ways

Thus, total number of password combinations = 5040 *  26⁴ * 10³ = 2.303159 * 10^{12 ways}

Hope this helps. Ask if you have any doubts.