Fundamental counting principle: If the first event could occur in m ways and the second event could occur in n ways, then the number of ways that the two events could occur in sequence is m*n

Solution Number of 4-letter strings

Each letter in the string has 5 possible values (a,b,c,d).

First letter: 5 ways

Second letter: 5 ways

Third letter: 5 ways

Fourth letter: 5 ways

Use the fundamental counting principle: \(\displaystyle{5}\cdot{5}\cdot{5}\cdot{5}={5}^{{4}}={625}\)

Number of 4-letter strings containing "aa" There are three possible positions for the string aa (that is, the string is either of the form aaxx, xaax or xxaa.

Each of the letters x in the string has 5 possible values (a,b,c,d,e).

Place aa: 3 ways First letter: 5 ways Second letter: 5 ways

Use the fundamental counting principle: \(\displaystyle{3}\cdot{5}\cdot{5}={75}\)

Number of 4-letter strings not containing "aa" There are 625 4-letter strings and there are 75 4-letter strings containing "aa", thus there are then 625-75=550 4-letter strings not containing "aa"

Solution Number of 4-letter strings

Each letter in the string has 5 possible values (a,b,c,d).

First letter: 5 ways

Second letter: 5 ways

Third letter: 5 ways

Fourth letter: 5 ways

Use the fundamental counting principle: \(\displaystyle{5}\cdot{5}\cdot{5}\cdot{5}={5}^{{4}}={625}\)

Number of 4-letter strings containing "aa" There are three possible positions for the string aa (that is, the string is either of the form aaxx, xaax or xxaa.

Each of the letters x in the string has 5 possible values (a,b,c,d,e).

Place aa: 3 ways First letter: 5 ways Second letter: 5 ways

Use the fundamental counting principle: \(\displaystyle{3}\cdot{5}\cdot{5}={75}\)

Number of 4-letter strings not containing "aa" There are 625 4-letter strings and there are 75 4-letter strings containing "aa", thus there are then 625-75=550 4-letter strings not containing "aa"