Number of common children of Mr. John and Ms. Bashu = 10 − ( x + x + 1) = 9 − 2 x Let N = The number of fights between children of different parents = (Total number of fights that can take place among all the children) − (The number of fights among the children of the same parents) =10C2−[xC2+x+1C2+9−2xC2]...(i) =45−[
x(x−1)
2
+
(x+1)(x)
2
+
(9−2x)(8−2x)
2
] =45−
1
2
[x2−x+x2+x+72−34x+4x2] =
397
12
−3(x−
17
6
)2 For N to be maximum, x must be
17
6
As x can not be in fraction, we take x = 3 (≈
17
6
) Thus, maximum value of N = 33 which is attained at x = 3.