Question

John has x children by his first wife. Mary has (x + 1) children by her first husband. They marry and have children of their own. The whole family has 24 children. Assuming the two children of same parents do not fight, then find the maximum possible number of fights that can take place –

• A. 192
• B. 189
• C. 190
• D. 191

Answer 1

Answer: (D)

191

Answer 2

Since the whole family has 24 children, those of John and Mary are;

24 – x – (x + 1)

i.e., (23 – 2x)

Now, F = Total number of fights.

= (number of fights when a John child fights a Mary child) + (number of fights when a John child fights a John Mary child) + (number of fights when a Mary child fights a John Mary child).

= x (x + 1) + x (23 – 2x) + (x + 1) (23 – 2x)

= 23 + 45x – 3x²

For maximum, $\frac{dF}{dx}$ = 0.

Ž 45 – 6x = 0

or x = 7.5

\ ƒ(x) is minimum when x = 7.5

But in this case fractional value is not possible.

The nearest integral values are x = 7 and x = 8.

In either case the total number of fights

= 23 + 45 × 7 – 3(7)² = 191.