6 different letters of an alphabet are given. Words with four letters

1.	6 different letters of an alphabet are given. Words with four letters are formed from these given letters. Determine the number of words which have at least one letter repeated. (a) 1296 (b) 996 (c) 936 (d) 360
Answer» (c) 936 The number of 4-letter words which can be formed from 6 letters when one or more of the letters is repeated = 6 × 6 × 6 × 6 = 1296 Number of 4-letter words which can be formed from the given 6 letters when none of the letters is repeated = Number of arrangements of 6 letters taken 4 at a time = ⁶P₄ = \(\frac{6!}{(6-4)!} = \frac{6!}{2!}\) = 6 × 5 × 4 × 3 = 360 ∴ Number of 4 letter words which have at least one of their letters repeated = 1296 – 360 = 936.

6 different letters of an alphabet are given. Words with four letters are formed from these given letters. Determine the number of words which have at least one letter repeated. (a) 1296 (b) 996 (c) 936 (d) 360

Answer»

(c) 936

The number of 4-letter words which can be formed from 6 letters when one or more of the letters is repeated = 6 × 6 × 6 × 6 = 1296

Number of 4-letter words which can be formed from the given 6 letters when none of the letters is repeated

= Number of arrangements of 6 letters taken 4 at a time

= ⁶P₄ = \(\frac{6!}{(6-4)!} = \frac{6!}{2!}\) = 6 × 5 × 4 × 3 = 360

∴ Number of 4 letter words which have at least one of their letters repeated = 1296 – 360 = 936.