Question

Permutations and combinations.

a.)A typesetter has before him 26 trays, one for each letter of the alphabet. Each tray contains

10 copies of the same letter. In how many ways can he form a three letter word that

requires at most two different letter?

b.)Determine the number of ways of forming words which use exactly two different letters.

Answer 1

a) For at most two different letters, we have the following two mutually exclusive cases:

Case 1: The first two letters are same. The third letter can be arbitrarily chosen from the remaining letters after first two are placed.

Number of ways = 260 x 9 x (260 - 2) = 603,720.

Case 2: The first two letters are different. The third letter must match either of the first two, so it must be one of the 18 letters remaining in the two trays used for the first two letters.

Number of ways = 260 x 250x 18 = 1,170,000.

Hence, number of ways to form a word using at most two different letters

= 603,720 + 1,170,000 = 1,773,720.

b) To calculate number of ways to form a word using exactly two different letters, we can just subtract from above, the number of ways of forming words using just one letter.

Number of ways of forming words using just one letter = 260 x 9 x 8 = 18,720

Hence, number of ways to form a word using exactly two different letters

= 1,773,720 - 18,720 = 1,755,000.