Question

How many different ways the letters of the word yogesh can be placed in 8 boxes?

Answer 1

Let's count cases based on how many cells of the middle row are occupied

All 44 cells. Then you have 22 remaining xx's and you should put 11 in the first row (22 options) and 11in the third row (another 22 options). So2×2=42×2=4ways.33 cells. You can occupy 33 cells in (43)=4(43)=4 ways. This leaves you with 33 more xx's. Choose now a row from rows 11 and 33 (22 ways to do that), put 22 of the xx's in that row and put the last xx in the remaining row (22 cells to choose from). So4×2×2=164×2×2=16ways.22 cells. You can occupy 22 cells in (42)=6(42)=6 ways. This leaves you with 44 more xx's and no other choice but to place them in the remaining cells. So6×1=66×1=6ways.

If you occupy less than 22 cells in the middle row, then there is not enough space for the remaining xx's, so that was all. Summing up

4+16+6=26