Answer to Question #149822 in Combinatorics | Number Theory for Wesam

Question

Compute the number of ways to write the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9 in the cells of a 3 by 3 grid such that
•each cell has exactly one number
,•each number goes in exactly one cell,
•the numbers in each row are increasing from left to right
,•the numbers in each column are increasing from top to bottom, and
•the numbers in the diagonal from the upper-right corner cell to the lower-left corner cell are increasing from upper-right to lower-left.

Answer 1

Denote by A, B, C - the numbers in the first row, by D, E, F - the numbers in the second row, and by G, H, I - the numbers in the third row (from left to right in each row).

A is the least number in the table, therefore, A = 1.

I is the greatest number in the table, therefore, I = 9.

There are 4 numbers (A, B, C, D) which are less than E, and there are 4 numbers (F, G, H, I) which are greater than E. Therefore, E = 5.

Furthermore, the set {B, C, D} = {2, 3, 4} and B < C. To determine the values of these variables it is sufficient to determine the value of D (total of 3 variants). Then B will be the minimal element of the set {2,3,4}\{D} and C will be the maximal element of this set.

Similarly, the set {F, G, H} = {6, 7, 8} and G < H. To determine the values of these variables it is sufficient to determine the value of F (total of 3 variants). Then G will be the minimal element of the set {6, 7, 8}\{F} and H will be the maximal element of this set.

Finally, the total number of ways to fill the table is equal to 3 x 3 = 9.