The Mississippi Problem

Actually there's a better way to get 53 ways. We need to find two empty space in the 11 spaces of the 4 rows in the middle. But those two space couldn't appear together in the second or the fourth row, therefore we have to minus 2 from the final result. So that's: 11C2 - 2= 53

Frst, we choose which 11 of the 13 boxes to fill with letters. This is equivalent to choosing two squares to be empty.

Label the rows A, B, C, D, E, F, from top to bottom. The two empty boxes are either in the same row, or they are in different rows. If they are in the same row, it must be row C or row E. There are $\binom{4}{2} = 6$ ways to choose two squares from row C, and $\binom{3}{2} = 3$ ways to choose two squares from row E.

Otherwise, the two empty squares are in different rows. We choose two rows from rows B, C, D, E, and then we choose a square from each row. This leads to $2 \cdot 4 + 2 \cdot 2 + 2 \cdot 3 + 4 \cdot 2 + 4 \cdot 3 + 2 \cdot 3 = 44$ possibilities.

So there are $6 + 3 + 44 = 53$ ways to choose two squares to be empty, so 53 ways to choose which 11 of the 13 boxes to fill. There are then $\frac{11}{4! 4! 2!} = 34650$ ways to arrange the letters of MISSISSIPPI, for a total of $53 \cdot 34650 = 1836450$ ways.