# How Many Words Can Be Formed by Taking 4 Letters at a Time from the Letters of the Word 'Moradabad'? - Mathematics

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 2C2 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 2C1 ways.
Now, from the remaining 5 letters, 2 letters can be chosen in 5C2 ways.
Thus, 2 alike letters and 2 different letters can be selected in 2Cx 5C2 = 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 6C4, 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

Concept: Combination
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 17 Combinations
Exercise 17.3 | Q 8 | Page 23