BDMO

Probability Level 3

There were 36 people in a party. Some of the peoples shook hand with each other. No two of them shook hands with each more than once. It was found that no two peoples with the same number of handshakes made, had shaken hands each other. Find the maximum number of handshakes at the party.

2334 5780 3487 4560 3885

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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1 solution

Munem Shahriar
Oct 14, 2017

Suppose that the number of peoples who shook hands with exactly i i other peoples is f ( i ) f(i) . Then, due to the given condition, f ( i ) ≤ 36 − i . f(i) \le 36 - i. Now, the total number of handshakes is 1 2 ∑ i = 0 35 i f ( i ) . \dfrac{1}{2} \sum_{i=0}^{35} if(i). So,

1 2 ∑ i = 0 35 i ⋅ f ( i ) ≤ 1 2 ∑ i = 0 35 i ( 36 − i ) = 3885 \frac{1}{2} \sum_{i=0}^{35}i \cdot f(i) \le \dfrac{1}{2} \sum_{i=0}^{35} i(36-i) = 3885

The maximum number of handshakes at the party is 3885 . \boxed{3885}.

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Something's not right here. For 36 people, the maximum number of handshakes occurs when every two people shake hands, which gives 36 ⋅ 35 2 = 630 \frac{36 \cdot 35}{2} = 630 handshakes. How can the answer be greater than this?

Jon Haussmann - 3 years, 1 month ago

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