Question

A team of 12 technicians is in a meeting. at the end of the meeting each person gives each other a high five. how many high fives are given

Answer 1

If n technicians give high five to all the other technicians, then the number of high fives each technician gives is (n-1), but each time a technician gives a high five, the other, the receiver of the action, also gives a high five; therefore the number of high fives will be: $\frac{n(n-1)}{2}$
Where n is the number of technicians.

Now, we can substitute the value to get out answer:
$\frac{12(12-1)}{2} = \frac{12(11)}{2} = \frac{132}{2} =66$

We can conclude that at the end of the meeting 66 high fives were given.

Answer 2

In this item, we are to determine the number of high fives given considering that there are 12 total number of participants in the meeting. We know for a fact that each high five will require 2 participants. We can determine the answer to this item by using the concept of Combination. 

Combination of 12 taken 2 can be expressed as 12C2 and can be solved using the equation,

       n = (12!)/(2!)(12 - 2)! 

This expression is equal to 66. Hence, the answer to this item is 66 high fives.