Question

Find the number of ways in which
(i) a selection of four letters can be made from the letters of the word 'PROPORTION'

Answer 1

As there are 10 letters in the word PROPORTION,
i.e., OOO, PP, RR, I, T and N.

We have to select four letters.

Case (I): 3 alike letters and 1 distinct letter:
There are 3 alike letters, OOO, which can be selected by 1 way. And Out of the 5 different letters, P, R, I, T and N, one can be selected by ${}^5{C}_1=5$ ways.

Case (II): 2 alike of one kind and 2 alike of other kind. There are 3 groups of two alike letters, so it can be selected by $3C_2=3$ ways.

Case (III): 2 alike letters and 2 different letters.
There are 3 pairs of two alike letters of which one pair can be selected in $3C_1$ ways.
Now, from the remaining 5 different letters, 2 different letters can be chosen by $5C_2$ ways.
So, 2 alike letters and 2 different letters can be selected by $3C_1\times {}^5{C}_2=3\times 10=30$ ways.

Case (IV) 4 different letters.
Here are 6 different letters. So, number of ways of selecting 4 letters
$={}^6{C}_4=\frac{6!}{4!2!}=\frac{6\times 5\times 4!}{4!2!}=15$

Hence, the total number of selections
$=5+3+30+15=53$