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

We have a tournament with two levels:

Junior level: 43 participants total

Senior level: 51 participants total

Each participant plays exactly one match against every other participant at their own level (no junior vs senior matches).

When we say "each pair plays one match," we're counting combinations, not arrangements. A match between person A and person B is the same as a match between person B and person A - it's just one match.

The combination formula is: ${{}}^{n}C_{2} = \dfrac{n(n-1)}{2}$

This tells us how many ways we can choose 2 people from n people to form pairs.

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Given: Girl vs girl matches in junior level = 153

Let $a$ = number of girls in junior section

Using the combination formula:

${{}}^{a}C_{2} = 153$

This means: $\dfrac{a(a-1)}{2} = 153$

$a(a-1) = 306$

We need two consecutive numbers whose product is 306.

Try $a = 18$: $18 \times 17 = 306$

Therefore: $a = 18$ girls in junior section

Since total juniors = 43:

Boys in junior section = $43 - 18 = 25$

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Given: Boy vs boy matches in senior level = 276

Let $b$ = number of boys in senior section

Using the combination formula:

${{}}^{b}C_{2} = 276$

This means: $\dfrac{b(b-1)}{2} = 276$

$b(b-1) = 552$

We need two consecutive numbers whose product is 552.

Try $b = 24$: $24 \times 23 = 552$

Therefore: $b = 24$ boys in senior section

Since total seniors = 51:

Girls in senior section = $51 - 24 = 27$

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Summary of participants:

Level	Boys	Girls	Total
Junior	25	18	43
Senior	24	27	51

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Boys can only play against girls within their own level.

Boy vs Girl matches = (Junior boys × Junior girls) + (Senior boys × Senior girls)

Junior level: $25 \times 18 = 450$ matches

Senior level: $24 \times 27 = 648$ matches

Total boy vs girl matches = $450 + 648 = 1098$

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Key insights:

When counting matches between pairs, we use combinations because order doesn't matter.

When solving $n(n-1) = k$, we look for two consecutive integers whose product equals $k$.

Participants only compete within their level, so we calculate matches separately for each level.

Answer: 1098