Question

A person buys eight packets of TIDE detergent. Each packet contains one coupon, which bears one of the letters of the word TIDE. If he shows all the letters of the word TIDE, he gets one free packet. If he gets exactly one free packet, then the number of different possible combinations of the coupons is:

• A. ${}^7{C}_3$
• B. ${}^7{C}_3-1$
• C. ${}^8{C}_3$
• D. ${}^7{C}_3-2$

Answer 1

The correct option is B ${}^7{C}_3-1$

Let $x_1,x_2,x_3,x_4$ be the number of times T, I, D, E letter appears on the coupon.
Then we must have :
$x_1+x_2+x_3+x_4=8\text{where}1\le x_1,x_2,x_3,x_4\le 8$
(as each letter must appear once).
Then the required number of combinations of coupons is equivalent to number of positive integral solutions of the above equation, which is further equivalent to number of ways of $8$ identical objects distributed among $4$ persons when each gets at least one objects, which is given by
${}^{8-1}{C}_{4-1}={}^7{C}_3$

But since he get exactly one packet means all the letters of the word TIDE should not occur again i.e. when $x_1=2,x_2=2,x_3=2,x_4=2$
So the required answer is ${}^7{C}_3-1$