How many different blackwhite colorings of the 3 times 3 gri

How many different black-white colorings of the 3 times 3 grid have exactly three black squares and six white squares? How many have exactly two black squares and seven white squares? Repeat the problem of counting the black-white colorings of the 3 times 3 grid, but do so for the 4 times 4 and the 5 times 5 grids. Generalize the previous problem to count the k-colorings of the n times n grid.

Solution

10. Total no. of cells = 3x3 = 9

Selecting exactly 3 black squares from 9 squares is ⁹C₃

Total no. of coloring possible = ⁹C₃ .⁶C₆ = 84

Selecting exactly 2 black squares from 9 squares is ⁹C₂

Total no. of coloring possible = ⁹C₂ .⁷C₇ = 36

11. Total no. of cells = 4x4 = 16

Selecting exactly 3 black squares from 16 squares is ¹⁶C₃

Total no. of coloring possible = ¹⁶C₃ .¹³C₁₃= 560

Selecting exactly 2 black squares from 16 squares is ¹⁶C₂

Total no. of coloring possible = ¹⁶C₂ .¹⁴C₁₄ = 120

Total no. of cells = 5x5 = 25

Selecting exactly 3 black squares from 25 squares is ²⁵C₃

Total no. of coloring possible = ²⁵C₃ .²²C₂₂ = 2300

Selecting exactly 2 black squares from 25 squares is ²⁵C₂

Total no. of coloring possible = ²⁵C₂ .²³C₂₃ = 300

12. if k no of black squares out of n squares,

Total no. of possible coloring = ⁿC_k.^n-kC_n-k