Explanation

Multiple approaches are possible here. One way is to imagine the scenario and count up the number of handshakes. How many hands does everyone need to shake?

There are 11 other people in the room, so the first person needs to shake hands 11 times. Now, move to the second person: how many hands must he shake? He has already shaken one hand, leaving him 10 others with whom to shake hands.

The third person will need to shake hands with 9 others, and so on. Therefore, there are a total of 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 handshakes. The fastest way to find the sum of a group of consecutive numbers is to take the average of the first and last terms and multiply it by the number of terms.

The average is \(\frac{11+1}{2}= 6\) and there are \(11 - 1 + 1 = 11\) terms (find the difference between the terms and “add one before you’re done”). The sum is \(6 \times 11 = 66\).

Another approach is to ask "In how many different ways can we select 2 people from 12 people?"
The idea here is that, for each unique selection of 2 people, we can get those people to shake hands.

The order in which we select the two people does not matter. For example, selecting Person A 1st and Person B 2nd is exactly the same as selecting Person B 1st and Person A 2nd.
Since the order in which we select the two people does not matter, we can use combinations.

We can select 2 people from 12 people in 12C2 ways.
12C2 = 66

Answer: C

RELATED VIDEO FROM OUR COURSE

Having just seen Brent's problem on finding the number of triangles in a circle (https://www.youtube.com/watch?v=OQpybhVoPms) I made a circle and thought of how many straight lines could connect the points on the circle. I solved as he showed above, using nCk.

I think the idea of the circle might help in similar problems where maybe they ask for how many squares could be formed, or pentagons, etc, or how many groups of 3 people shaking hands (which I assume would be the same as the number of triangles).

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