What is the formula for the handshake problem?

Question

anonymous · Answer

how many handshakes required between $n$ different people?

ranga · Answer

I have answered your previous two questions.  nC2 is the formula.

amberlykhan · Answer

6 people and everyone shakes with each other once.

anonymous · Answer

indeed, it's just the number of distinct pairs of people you can make i.e. $\dbinom{n}2=\dfrac{n!}{2!(n-2)!}$

jdoe0001 · Answer

the "secret" handshake =)

ranga · Answer

In how many ways can two people be chosen out of 6?  nC2 ways.

amberlykhan · Answer

So can you fill the variables so I can see?

ranga · Answer

6C2 ways

anonymous · Answer

notice$$\frac{n!}{2!(n-2)!}=\frac{n(n-1)}2$$ ways... just let $n=6$

anonymous · Answer

it's trivial to show that if we cared about who initiated each handshake, then each of the $n$ people would need to initiate $n-1$ handshakes so that answer would be $n(n-1)$ (by the fundamental principle of counting). since we don't care who initiates each handshake, we're actually counting exactly *twice* as many handshakes as necessary hence $n(n-1)/2$ is our final answer.

for $n=6$ we have 6 people who each initiates handshakes with 5 others so $6\times5=30$ handshakes. but this means that A shakes B's hand and B shakes A's hand so we're actually counting twice as many handshakes as needed so we divide by $2$ like above:

$$30/2=15$$

amberlykhan · Answer

thx!