2020 TJC P2 Q7

The diagram shows a \(3\times 3\) grid of \(9\) cells, numbered \(1\) to \(9\).

The nine letters T, T, T, T, J, C, C, C, C are to be arranged in the grid, with each cell occupied by one letter.

Find the number of ways to arrange the nine letters if

1. there are no restrictions,[2]
2. the cells at the four corners of the grid must each be occupied by a ‘T’.[1]

Two letters are considered adjacent to each other if they occupy cells that are above, below, to the left or to the right of each other.

Find the number of ways to arrange the nine letters if

3. all ‘T’s must not be adjacent to another ‘T’ and all ‘C’s must not be adjacent to another ‘C’,[2]
4. at least two ‘T’s must be adjacent and at least two ‘C’s must be adjacent.[3]