JEE MAIN - Mathematics (2019 - 12th April Evening Slot - No. 16)
A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can
randomly be selected from this group such that there is at least one boy and at least one girl in each team, is
1750, then n is equal to :
24
25
27
28
Explanation
Given that 5 Boy, n girls.
(1B, 2G) + (2B, 1G)
$${}^5{C_1}.{}^n{C_2} + {}^5{C_2}.{}^n{C_1} = 1750$$
$$ \Rightarrow 5.{{n\left( {n - 1} \right)} \over 2} + 10.n = 1750$$
$$ \Rightarrow {{n\left( {n - 1} \right)} \over 2} + 2n = 350$$
$$ \Rightarrow {n^2} - n + 4n = 700$$
$$ \Rightarrow {n^2} + 3n - 700 = 0$$
$$ \Rightarrow (n + 28)(n - 25) = 0$$
$$ \Rightarrow n = 25, -28$$
(1B, 2G) + (2B, 1G)
$${}^5{C_1}.{}^n{C_2} + {}^5{C_2}.{}^n{C_1} = 1750$$
$$ \Rightarrow 5.{{n\left( {n - 1} \right)} \over 2} + 10.n = 1750$$
$$ \Rightarrow {{n\left( {n - 1} \right)} \over 2} + 2n = 350$$
$$ \Rightarrow {n^2} - n + 4n = 700$$
$$ \Rightarrow {n^2} + 3n - 700 = 0$$
$$ \Rightarrow (n + 28)(n - 25) = 0$$
$$ \Rightarrow n = 25, -28$$
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