(Q). In how many ways 2 + and 2 – signs are filled into 4×4 cell where each cell can contain maximum one character such that each row & column can not contain same sign ?
Solution:-
1ˢᵗ ‘+’ sign can be arrranged in ¹⁶C₁ = 16 ways
2ⁿᵈ ‘+’ sign can be arranged in ⁹C₁ = 9 ways
Since both ‘+’ signs are indistinguishable so both ‘+’sign can be arranged in = 72 ways
similarly both ‘-‘ sign can be arranged in = 72 ways
Hence 2 ‘+’ & 2 ‘-‘ sign can be arranged in 72×72 ways.
From this valuewe have to exclude the following cases to get the final answer:-
case (i): two group (+, -) & (+, -) can be arranged in = 72 ways
case(ii):
Hence symbol of case (ii) according to condition can be arranged in 16×9×8 ways
Hence required Answer is:
= 72×72 – 72 – 16×9×8
=3960 Answer