Question

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

Answer 1

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