combinatorics problem PART 1
Probability
Level
2
What is the number of ways in which one can colour the square of a 4 x 4 chessboard with colours red and blue such that each row as well as each column has exactly two red squares and blue squares ? The answer is N what is the value of N/15
3
66
6
2
4
8
5
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1 solution
First row can be filled by 4C2 ways = 6 ways.
Case-I Second row is filled same as first row
here second row is filled by one way
3rd row is filled by one way
4th row is filled by one way
Total ways in Case-I equals to 4C1 × 1 × 1 × 1 = 6 ways
Exactly 1 R & 1 B is interchanged in second row in comparision to 1st row
here second row is filled by 2 × 2 way
3rd row is filled by two way
4th row is filled by one way
Total ways in Case-II equals to 4C1 × 2 × 2 × 2 × 1 = 48 ways
Both R and B is replaces by other in second row as compared to 1st row
here second row is filled by 1 way
3rd row is filled by 4C2 way
4th row is filled by one way
Total ways in 3rd Case equals to 4C2 × 1 × 6 × 1 = 36 ways
Total ways of all cases equals to 90 ways