Let's Party

Probability Level 4

A party is attended by twenty people. In any subset of four people, there is at least one person who knows the other three. Suppose there are three people in the party who do not know each other. How many people in the party know everyone?

NOTE : We assume that if $X$ knows $Y$ , then $Y$ knows $X$ .

17 18 Cannot be determined from the given data. 16

1 solution

Shinya Kogami
May 29, 2018

Let us label the people $\{1,2\cdots 20\}$ . Let us suppose that $\{1,2,3\}$ do not know each other.

Consider $\{1,2,3,x\}$ where $x \in \{4,5\cdots ,20\}$ . This shows that each one of $\{4,5\cdots 20\}$ knows $1,2,3$ .

Now consider $\{x,1,2,y\}$ where $x \ne y$ and $x,y \in \{4,5\cdots\}$ . This shows that $x$ and $y$ know each other.

Hence , each one of $4,5\cdots 20$ knows everyone.

Thus, $\boxed{17}$ people know everyone.

Why not include 3 in the group...i mean 3,4,5,.....,20....since every one from 4 to 20 must know 3??

rajdeep brahma - 3 years ago

Let's Party

1 solution

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