Question

Find the number of ways in which : (a) a selection

Answer 1

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

(a) 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

\[{}^5 C_1\]= 5 ways.

(ii) There are 3 sets of two alike letters, which can be selected in ³C₂ = 3 ways.
(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.
(iv) There are 6 different letters.
Number of ways of selecting 4 letters = ⁶C₄ = 15
∴ Total number of ways = 5+ 3 + 30 + 15 = 53