Question

Let a₁, a₂, ...., a_n be n non-zero real numbers, of which p are positive and remaining are negative. The number of ordered pairs (j, k), j < k, for which a_j a_k is positive, is 55. Similarly, the number of ordered pairs (j, k), j < k, for which a_j a_k is negative, is 50. Then, the value of p² + (n - p)² is

(a) 629

(b) 325

(c) 125

(d) 221

Answer 1

Answer is : (c) 125

Let p are positive number from a₁, a₂, ...., a_n

∴ n - p are negative number.

Given a_j a_k, is positive j < k and a_j a_k = 55

a_j a_k is positive.

∴ a_j and a_k are both positive or negative.

and a_j a_k is negative j < k

a_j a_k = 50

any one of a_j and a_k are positive: