Let a₁, a₂, a_ n be n nonzero 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 a_ k is negative is 50 . Then the value of p²+(n-p)² is

Step-by-step solution

^ P C ₂+ ^ n-p C ₂=55 p ( p -1) 2 + ( n - p )( n - p -1) 2 =55 .........(i) Also, p(n-p)=50 .........(ii) Put in (i) array l p(p-1)+ 50 p ( 50 p -1 )=110 p²-p+ ( 50 p )²- 50 p =110 p²-p+ ( 50 p )²- 50 p =110 (p+ 50 p )²-100- (p+ 50 p )=110 t²-t-210=0 t=15 or -14 (not true) p+ 50 p =15 array To find p²+(n-p)²=p²+ ( 50 p )² array l = (p+ 50 p )²-100 =125 (using (iii)) array