How many words can be formed by taking 4 letters at a time from the letters of the word 'MORADABAD'?

#### Solution

There are 9 letters in the word MORADABAD, namely AAA, DD, M, R, B and O.

The four-letter word may consists of

(i) 3 alike letters and 1 distinct letter

(ii) 2 alike letters of one kind and 2 alike letters of the other kind

(iii) 2 alike letters and 2 distinct letters

(iv) all different letters

(i) 3 alike letters and 1 distinct letter:

There is one set of three alike letters, AAA, which can be selected in one way.

Out of the 5 different letters D, M, R, B and O, one can be selected in \[{}^5 C_1\]ways.

These four letters can be arranged in\[\frac{4!}{3! 1!}\]ways.

∴ Total number of ways = \[{}^5 C_1 \times \frac{4!}{3! 1!} = 20\]

(ii) There are two sets of two alike letters, which can be selected in ^{2}C_{2} ways.

Now, the letters of each group can be arranged in \[\frac{4!}{2! 2!}\]ways.

∴ Total number of ways =\[{}^2 C_2 \times \frac{4!}{2! 2!} = 6\]

(iii) There is only one set of two alike letters, which can be selected in ^{2}C_{1} ways.

Now, from the remaining 5 letters, 2 letters can be chosen in ^{5}C_{2} ways.

Thus, 2 alike letters and 2 different letters can be selected in ^{2}C_{1 }x ^{5}C_{2} = 20 ways.

Now, the letters of each group can be arranged in \[\frac{4!}{2!}\]

∴ Total number of ways = \[20 \times \frac{4!}{2!} = 240\]

(iv) There are 6 different letters A, D, M,B, O and R.

So, the number of ways of selecting 4 letters is ^{6}C_{4}, i.e. 15, and these letters can be arranged in 4! ways.

∴ Total number of ways = 15 x 4! = 360

∴ Total number of ways = 20 + 6 + 240 + 360 = 626