Question

Combinatorics Chessboard (8x8 grid), show all work.

a. How many ways are there to put 8 (identical) pennies on a chessboard, so no two share a row, and no two share a column?

b. How many ways are there to put 5 pennies on a chessboard, with the same restriction?

c. How many ways are there to put 3 pennies and 5 nickles on the board, with the same restriction - no two coins share a row, no two coins share a column?

Answer 1

a.) 1st penny have 64 options. But as we need the second should not be in the same row or column of the one already placed thus, there are 49 options left for the second penny. Similarly going on the third have 36 options and so on. Thus the number of ways of placing 8 pennies on the chessboard is given by 64 * 49 * 36 * 25 * 16 * 9 * 4 * 1 = 8!*8! . Now as the pennies are identical we have to divide by 8!, Thus the answer is 8!

b). Proceeding in a similar fashion the number of ways here is 64 * 49 * 36 * 25 * 16 and divide by 5!, i.e.

$\frac{64*49*36*25*16}{5!}$

c). Here the number of ways is similar to the 1st part is 8! 8! and here 5 nickels and 3 pennies are identical hence, the answer is $\frac{8!*8!}{5!*3!}$