How many different 55-letter words can be made a. if the first letter must be Upper K or Upper XK or X and no letter may be repeated? b. if repeats are allowed (but the first letter is Upper K or Upper XK or X)? c. How many of the 55-letter words (starting with Upper K or Upper XK or X) with no repeats end in Upper JJ?

(a) The number of 5-letter word that can be formed (without repetition) such that the first letter is K or X is 607,200.

(b) The number of 5-letter word that can be formed (with repetition) such that the first letter is K or X is 913,952.

(c) The number of 5-letter word that can be formed (without repetition) such that the first letter is K or X and the last is J is 24,288.

Explanation:

In this case we need to form a 5 letter word.

(a)

The condition applied is: 1st letter is K or X.

The word will of the form: K _ _ _ _ or X _ _ _ _

There are 2 ways to fill the first space, i.e. with either K or X.

Repetition is not allowed.

The remaining 4 spaces can be filled in (25 × 24 × 23 × 22) ways.

Then the total number of 5-letter word that can be formed (without repetition) such that the first letter is K or X is:

Number of 5-letter word = 2 × 25 × 24 × 23 × 22 = 607200

Thus, the number of 5-letter word that can be formed (without repetition) such that the first letter is K or X is 607,200.

(b)

In case the repetition are allowed the number of 5-letter word that can be formed (without repetition) such that the first letter is K or X is:

Number of 5-letter word = 2 × 26 × 26 × 26 × 26 = 913952

Thus, the number of 5-letter word that can be formed (with repetition) such that the first letter is K or X is 913,952.

(c)

The condition applied is: 1st letter is K or X and the last letter is J.

The word will of the form: K _ _ _ J or X _ _ _ J

There are 2 ways to fill the first space, i.e. with either K or X, and the last with J.

Repetition is not allowed.

The remaining 3 spaces can be filled in (24 × 23 × 22) ways.

Then the total number of 5-letter word that can be formed (without repetition) such that the first letter is K or X and the last is J is:

Number of 5-letter word = 2 × 24 × 23 × 22 = 24288

Thus, the number of 5-letter word that can be formed (without repetition) such that the first letter is K or X and the last is J is 24,288.

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