SAT1000 - P937

Probability Level 3

Six classmates are exchanging souvenirs among each other in the graduation party after Gaokao. For each pair of two students, they can exchange once at most. For each round of exchange, the two students send each other a souvenir.

Given that these 6 6 classmates have performed 13 13 rounds of exchange, what is the number of classmates that have received exactly 4 4 souvenirs?

A . 1 or 3 A.\ \textup{1 or 3}

B . 1 or 4 B.\ \textup{1 or 4}

C . 2 or 3 C.\ \textup{2 or 3}

D . 2 or 4 D.\ \textup{2 or 4}

Have a look at my problem set: SAT 1000 problems

A A D D B B C C

Alice Smith by Alice Smith , 20, China

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2 solutions

Jeff Giff
Jul 18, 2020

This is like joining 6 dots with 13 lines and seeing how many dots are connected to only four lines using brute force. It took me quite a long time to find the second possibility :P

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Adrian Klaeger
Jul 20, 2020

The complete graph with 6 vertices has 15 edges, with each vertex having degree 5. The question asks about 13 edges, so we remove two from the complete graph.

We can either remove two disjoint edges, which reduces the degree of 4 different vertices to 4; or we can remove two edges that share an endpoint, which reduces the degree of the shared vertex to 3 and those of the other two endpoints to 4.

The number of vertices with degree 4 are therefore 2 or 4.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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