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CAT 2018 Slot 2 — QA Question 29

Forming a committeeEasy

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

Answer & solution

Correct answer: 1098

Answer: 1098

Solution

Easy

In a group of nn players where every pair plays once, the number of matches is (n2)=n(n−1)2\binom{n}{2}=\tfrac{n(n-1)}{2}. Use the given girl–girl and boy–boy match counts to recover how many girls and boys there are at each level, then count cross matches as (boys)×\times(girls) within each level.

1

Juniors: find the number of girls. Girl–girl matches =(g2)=153=\binom{g}{2}=153.

g(g−1)2=153⇒ g(g−1)=306=18⋅17⇒ g=18(junior girls)⇒ junior boys=43−18=25\begin{aligned} &\frac{g(g-1)}{2} = 153\\ &\Rightarrow\ g(g-1) = 306 = 18\cdot 17\\ &\Rightarrow\ g = 18 \quad\text{(junior girls)}\\ &\Rightarrow\ \text{junior boys} = 43 - 18 = 25 \end{aligned}
2

Seniors: find the number of boys. Boy–boy matches =(b2)=276=\binom{b}{2}=276.

b(b−1)2=276⇒ b(b−1)=552=24⋅23⇒ b=24(senior boys)⇒ senior girls=51−24=27\begin{aligned} &\frac{b(b-1)}{2} = 276\\ &\Rightarrow\ b(b-1) = 552 = 24\cdot 23\\ &\Rightarrow\ b = 24 \quad\text{(senior boys)}\\ &\Rightarrow\ \text{senior girls} = 51 - 24 = 27 \end{aligned}
3

Count boy–girl matches. No junior plays a senior, so cross matches occur only within each level: (boys)×\times(girls) per level.

juniors: 25×18=450seniors: 24×27=648⇒ total=450+648=1098\begin{aligned} &\text{juniors: } 25\times 18 = 450\\ &\text{seniors: } 24\times 27 = 648\\ &\Rightarrow\ \text{total} = 450 + 648 = 1098 \end{aligned}
1098\textbf{1098}

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CAT 2018 Slot 2 QA Q29: In a tournament, there are 43 junior level and 51 senior level participants. Each pair of juniors play one mat — Solution | TheCATExam