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Question

A team of three persons with at least one boy and atleast one girl is to be formed from $$5$$ boys and $$n$$ girls. If the number of sum teams is $$1750$$, then the value of $$n$$ is

A
$$25$$
B
$$24$$
C
$$27$$
D
$$28$$
Solution
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Correct option is D. $$25$$
Given $$5$$ boys, $$n$$ girls
$$(1B,2G)+(2B,1G)$$
$${ _{ }^{ 5 }{ C } }_{ 1 }.{ _{ }^{ n }{ C } }_{ 2 }+{ _{ }^{ 5 }{ C } }_{ 2 }.{ _{ }^{ n }{ C } }_{ 1 }=1750\Rightarrow 5.\cfrac { n(n-1) }{ 2 } +10.n=1750\Rightarrow \cfrac { n(n-1) }{ 2 } +2n=350\Rightarrow { n }^{ 2 }-n+4n=700$$
$${ n }^{ 2 }+3n-700=0\Rightarrow (n+28)(n-25)=0\Rightarrow n=25,-28$$
n $$\neq -28$$ as number of teams cannot be negative

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