# Find the Number of Ways in Which : (B) an Arrangement, of Four Letters Can Be Made from the Letters of the Word 'Proportion'. - Mathematics

Find the number of ways in which : (b) an arrangement, of four letters can be made from the letters of the word 'PROPORTION'.

#### Solution

There are 10 letters in the word PROPORTION, namely OOO, PP, RR, I, T and N.

(b) 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 second kind
(iii) 2 alike letters and 2 distinct letters
(iv) all distinct letters

Now, we shall discuss these four cases one by one.
(i) 3 alike letters and 1 distinct letter:
There is one set of three alike letters, OOO, which can be selected in one way.
Out of the 5 different letters, P, R, I, T and N, 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 3 sets of two alike letters, which can be selected in 3C2 ways.
Now, the letters of each group can be arranged in$\frac{4!}{2! 2!}$ways.
∴ Total number of ways =${}^3 C_2 \times \frac{4!}{2! 2!} = 18$

(iii) There are three sets of two alike letters, which can be selected in 3C1 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 3C1 x 5C2 = 30 ways.
Now, the letters of each group can be arranged in $\frac{4!}{2!}$ways.
∴ Total number of ways = $30 \times \frac{4!}{2!} = 360$

(iv) There are 6 different letters.
So, the number of ways of selecting 4 letters is 6C4 = 15 and these letters can be arranged in 4! ways.
∴ Total number of ways = 15 x  4! = 360

∴ Total number of ways = 20 + 18 + 360 + 360 = 758

Concept: Combination
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 17 Combinations
Exercise 17.3 | Q 7.2 | Page 23