We have a tournament with two levels: $$\newline$$ Junior level: 43 participants total $$\newline$$ Senior level: 51 participants total $$\newline$$ Each participant plays exactly one match against every other participant at their own level (no junior vs senior matches). $$\newline$$ 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. $$\newline$$ The combination formula is: ${{}}^{n}C_{2} = \dfrac{n(n-1)}{2}$ $$\newline$$ This tells us how many ways we can choose 2 people from n people to form pairs. $$\newline$$

$$\newline$$ Given: Girl vs girl matches in junior level = 153 $$\newline$$ Let $a$ = number of girls in junior section $$\newline$$ Using the combination formula: $\newline$ ${{}}^{a}C_{2} = 153$ $$\newline$$ This means: $\dfrac{a(a-1)}{2} = 153$ $$\newline$$ $a(a-1) = 306$ $$\newline$$ We need two consecutive numbers whose product is 306. $$\newline$$ Try $a = 18$: $18 \times 17 = 306$ $$\newline$$ Therefore: $a = 18$ girls in junior section $$\newline$$ Since total juniors = 43: $\newline$ Boys in junior section = $43 - 18 = 25$ $$\newline$$

$$\newline$$ Given: Boy vs boy matches in senior level = 276 $$\newline$$ Let $b$ = number of boys in senior section $$\newline$$ Using the combination formula: $\newline$ ${{}}^{b}C_{2} = 276$ $$\newline$$ This means: $\dfrac{b(b-1)}{2} = 276$ $$\newline$$ $b(b-1) = 552$ $$\newline$$ We need two consecutive numbers whose product is 552. $$\newline$$ Try $b = 24$: $24 \times 23 = 552$ $$\newline$$ Therefore: $b = 24$ boys in senior section $$\newline$$ Since total seniors = 51: $\newline$ Girls in senior section = $51 - 24 = 27$ $$\newline$$

$$\newline$$ Summary of participants: $$\newline$$

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

$$\newline$$

$$\newline$$ Boys can only play against girls within their own level. $$\newline$$ Boy vs Girl matches = (Junior boys × Junior girls) + (Senior boys × Senior girls) $$\newline$$ Junior level: $25 \times 18 = 450$ matches $$\newline$$ Senior level: $24 \times 27 = 648$ matches $$\newline$$ Total boy vs girl matches = $450 + 648 = 1098$ $$\newline$$

$$\newline$$ Key insights: $$\newline$$ When counting matches between pairs, we use combinations because order doesn't matter. $$\newline$$ When solving $n(n-1) = k$, we look for two consecutive integers whose product equals $k$. $$\newline$$ Participants only compete within their level, so we calculate matches separately for each level. $$\newline$$ Answer: 1098

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