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There are three teams x,x+1 and y childr...

There are three teams `x`,`x+1` and `y` childrens and total number of childrens in the teams is `24`. if two childrens of the same team do not fight, then

A

(a) maximum number of fights is 190

B

(b) maximum number of fights is 191

C

(c) maximum number of fights occur when x=7

D

(d) maximum number of fights occur when x=8

Text Solution

Verified by Experts

The correct Answer is:
B, C, D
`becausex+x+1+y=24`
`impliesy=23-2x`
Let N=Total number of fights subject to the condition that any two children of one tem do not fight.
`thereforeN=2.^(24)C_(2)-(.^(x)C_(2)+.^(x+1)C_(2)+.^(y)C_(2))`
`=.^(24)C_(2)-(.^(x)C_(2)+.^(x+1)C_(2)+.^(23-2x)C_(2))`
`=23-3x^(2)+45x`
`therefore(dN)/(dx)=0-6x+45`
for maximum or minimum, put `(dN)/(dx)=0impliex=7.5`
`impliesx=7" "[because x in I]`
Now, `(d^(2)N)/(dx^(2)) lt0`
`therefore` N will be maximum when x=7
and N=23-`3(7)^(2)+45xx7=191`
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