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In a tournament, there are $43$ junior level and $51$ senior level participants. Each pair of juniors play one match. Each pair of seniors play one match. There is no junior versus senior match. The number of girl versus girl matches in junior level is $153$, while the number of boy versus boy matches in senior level is $276$. The number of matches a boy plays against a girl is _________

Given that, in a tournament, there are $43$ junior level and $51$ senior-level participants ( Boys + Girls).

Let the number of girls in junior-level be $\text{‘G’}.$

And, the number of boys in senior-level be $\text{‘B’}.$

The number of girl Vs girl matches in junior-level $= 153$

$\Rightarrow \;^{\text{G}}c_{2} = 153$

$\Rightarrow \frac{\text{G!}}{(\text{G-2})! \; 2!} = 153$

$\Rightarrow \frac{\text{G (G-1) (G-2)!}} { \text{(G-2)}! \; 2!} = 153$

$\Rightarrow \text{G (G-1)} = 306$

$\Rightarrow \text{G}^{2} – \text{G} – 306 = 0$

$\Rightarrow \text{G}^{2} – 18 \text{G} + 17 \text{G} – 306 = 0$

$\Rightarrow \text{G(G-18)} + 17 \text{(G-18)} = 0$

$\Rightarrow \text{(G-18)(G+17)} = 0$

$\Rightarrow \boxed{ \text{G} = 18, \; – 17}$

Thus, the number of girls in junior level $\boxed {\text{G}= 18}$

So, the number of boys in junior level $= 43 – 18 = 25.$

The number of matches played between a boy and a girl $= 25 \times 18 = 450$

The number of boy Vs boy matches in senior-level $= 276$

$\Rightarrow \;^{ \text{B}}c_{2} = 276$

$\Rightarrow \frac{\text{B}!}{(\text{B-2})! \; 2!} = 276$

$\Rightarrow \frac{\text{B(B-1)(B-2)!}} {(\text{B} – 2)! \; 2!} = 276$

$\Rightarrow \text{B}^{2} – \text{B} = 552$

$\Rightarrow \text{B}^{2} – \text{B} – 552 = 0$

$\Rightarrow \text{B}^{2} – 24\text{B} + 23 \text{B} – 552 = 0$

$\Rightarrow \text{B(B-24)} + 23 \text{(B-24)} = 0$

$\Rightarrow \text{(B-24)(B+23)} = 0$

$\Rightarrow \boxed{\text{B} = 24, \; -23 }$

Thus, the number of boys in senior level $\boxed{ \text{B} = 24}$

So, the number of girls in senior level $= 51 – 24 = 27.$

The number of matches played between a boy and a girl $= 27 \times 24 = 648.$

$\therefore$ The number of matches a boy plays against a girl $= 450 + 648 = 1098.$

Correct Answer $:1098$

$\textbf{PS:}$ Among a group of $n$ person, number of matches played between them $= \;^{n}c_{2}$

$\quad = \frac{n!}{(n-2)! \; 2 \times1}$

$\quad = \frac{n(n-1)(n-2)!}{(n-2)! \; 2 \times 1}$

$\quad = \frac{n(n-1)}{2}$
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