Solution — 2021 Question 7 Permutations & Combinations

We take out the letters P and G and rearrange the rest first $(7! ways)$

inserting letters to have 5 in between

We now try to insert P and G such that there are $5$ letters in between them. There are $3$ possible ways

Finally, there are $2!$ ways to arrange P and G

\begin{matrix} Number of arrangements \\ = 7! \times 3 \times 2! \\ = 30240 ∎ \end{matrix}