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2024 PEA Q12

In a chess tournament, there are both boys and girls. Each player plays with another player exactly once. If there are 4545 games in total and exactly 1515 of them feature only boys, how many games feature a boy and a girl?

Reveal answer and solution

Answer

D

Solution

  1. 1

    Let bb and gg be the number of boys and girls. Then (b2)=15⇒b(b−1)=30⇒b=6\binom{b}{2}=15\Rightarrow b(b-1)=30\Rightarrow b=6. Also (b+g2)=45⇒(b+g)(b+g−1)=90⇒b+g=10⇒g=4\binom{b+g}{2}=45\Rightarrow (b+g)(b+g-1)=90\Rightarrow b+g=10\Rightarrow g=4.

  2. 2

    Boy--girl games =b⋅g=6⋅4=24=b\cdot g=6\cdot 4=24.

Answer structure / marking notes

No major trap beyond standard calculation care.

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Content note

Imported from public/resources/isi/msqe/solutions/pea/2024/ISI_MSQE_PEA_2024_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.