A product code is formed from the symbols Q, R, T, and Z. The code consists of four symbols arranged one after the other, for example QQRT. How many different product codes contain the letter T?

Question

Answers (its not A,B,C, or F)

a. 64

b. 6

c. 27

d. 192

e. 175

f. 108

Answer 1

the letter T can appear in the first position ONLY. In that case, there are 3^3=27 possible combinations.

The same is true for the letter T appearing in ONLY the second position...

and the 3rd ONLY...

and the 4th ONLY

so the subtotal thus far is 4*27 = 108

Now for the letter T appearing in 2 of the 4 letter T, there are (4 choose 2) possible ways to arrange this:

(4 choose 2) = 4*3/2 = 6, namely

TTxx

TxTx

TxxT

xTTx

xTxT

xxTT

where x=Q,R, or Z

so we have 3^2*6 = 9 *6 = 54 possible combinations for the letter T appearing twice.

This brings the total up to 108+54 = 162

Now for 3 appearances of the letter T, the template looks like (4 choose 3) = 4

TTTx

TTxT

TxTT

xTTT

So there are exactly 3*4=12 ways to pull this off, which brings the total up to 162+12 = 174.

Then finally we have the case of letter T appearing in all four positions.

The final answer is 175