How many different product codes contain exactly one Q?

Question

Assume that product codes are formed from the letters W, T, S, V, Y, X, and Q, and consist of 4 not necessarily distinct letters arranged one after the other. For example, WWVQ is a product code.

How many different product codes contain exactly one Q?



Answer 1

W, T, S, V, Y, X, Q

All of the possible codes with Q are as follows:

QQQQ = 1 possible code with 4 Qs

QQQ(W,T,S,V,Y,X) = 6 x 1

QQ, (W,T,S,V,Y,X), Q = 6 x 1 4(6) = 24 possible codes with 3 Qs

Q, (W,T,S,V,Y,X), QQ = 6 x 1

(W,T,S,V,Y,X), QQQ = 6 x 1

QQ (W,T,S,V,Y,X)(W,T,S,V,Y,X) = 6 x 6

Q (W,T,S,V,Y,X) (W,T,S,V,Y,X) Q = 6 x 6 3(36) = 108 possible codes with 2 Qs

(W,T,S,V,Y,X) (W,T,S,V,Y,X) QQ = 6 x 6

Q (W,T,S,V,Y,X)(W,T,S,V,Y,X)(W,T,S,V,Y,X) = 6 x 6 x 6 = 216

(W,T,S,V,Y,X) Q (W,T,S,V,Y,X)(W,T,S,V,Y,X) = 6 x 6 x 6 = 216

(W,T,S,V,Y,X)(W,T,S,V,Y,X) Q (W,T,S,V,Y,X) = 6 x 6 x 6 = 216

(W,T,S,V,Y,X)(W,T,S,V,Y,X)(W,T,S,V,Y,X) Q = 6 x 6 x 6 = 216

216 + 216 + 216 + 216 = 4(216) = 864 possible codes with 1 Q