Interesting Fact of Handshake Count

Suppose we fill Yankee Stadium with 50,000 people and ask them to spend the day shaking hands with one another.

Prove that, at the end of the day, at least two participants will have shaken hands with the same number of people.

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Proving Interesting Fact of Handshake Count

What was that fact?

Let's contradict the given fact & assume no 2 people have same number of handshakes. In that case, the most gregarious person would have 49999 handshakes & next gregarious person would shake hands with 49998 people and so on. 

This way, the shyest person should have 0 shake hands. But the most gregarious guy must have had handshake with this shyest guy as his count of 49999 also includes this shyest guy. So this is impossible case.

Hence, at least 2 participants would have shaken hands with the same number of people.

 
In other way, the most shyest participant would have 1 handshake, next shyer guy would have 2 & so on. The more gregarious would have 49999 handshakes that includes the shake hand with the most gregarious person. Now, most gregarious person is bound to have 49999 handshakes as he/she can't have 50000 as there are only 50000 people in the stadium including himself/herself. 

That's why, at least 2 participants would have shaken hands with the same number of people.

Count The Number of People From Handshakes

At a party, everyone shook hands with everybody else. There were 66 handshakes.
How many people were at the party?

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Getting Count of Number of Peoples From Handshakes

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Let's suppose that there are 'n' number of people in the party.

The first person will shake hand with (n-1) people, the second person will shake hand with (n-2) people, the third will shake hand with (n-3) people.

In this way, (n-1) th person will shake hand with n-(n-1) = 1 person i.e. last person.

Adding all the number of handshakes,

(n-1) + (n-2) + (n-3) + ..... + 3 + 2 + 1 = n[(n-1)/2]

But total handshakes given are - 66

n[(n-1)/2] = 66

n(n-1) = 132

n^2 - n - 132 = 0

(n-12)(n+11) = 0

n = 12 or n = -11

Since number of people can't be negative, n = 12.

Hence there are 12 people in the party.