can you please help us solve this maths question?

Question

we wish to make a string of letters which contains every possible three letter combinations of P's and Q's somewhere(that is, it mustcointain all these: PPP,PPQ,PQP,PQQ, QPP, QPQ,QQP and QQQ). An example of such a string,of length 18 is, PPPPQPQQPPQPQQPQQQ
What is the length of the shortest such string?

A:8

B:10

C:12

D:16

E:18

please show how you worked it out.

Answer 1

qqqpppqpqq

B 10

i used guessed and check

Answer 2

First we agree there are 8 combinations, so the best we can do is if we overlap the last two letters of one combination with the next. That would be 8 overlaps plus the starting and ending letters, for a total of 10.

But to prove this we have to find a way to overlap all 8 combinations. The way I did this was to start with PPP. The next letter will be the opposite of the first letter in the sequence of 3 letters. So PPP will go to PP+Q.
PPQ will go to PQ+P
PQP will go to QP+Q
etc.

The full sequence is:
PPP --> PPQ --> PQQ --> QQQ --> QQP --> QPQ --> PQP --> QPP

(Notice how this returns to PPP next, so we have completed the circle).

Anyway putting this together you get:
PPPQQQPQPP

(Also, you can create different arrangements by cycling these letters, or reversing the whole string, but they are all 10 letters long. That is the minimum.)

So the answer is:
B: 10.

Answer 3

ppp,ppq,pqp,pqq,qpq,qqp,qqq,qpp so...

qqpppqpqqq
so 10 is my answer, mostly because this is giving me a headache

Answer 4

Answer= A.8 (solved it on a Microsoft word document)

Answer 5

answer = FAHQ