Question

A curious patterns occurs in a group of people who all shake hands with one another. It turns out that you can predict the number of handshakes that will occur if you know the number of people. If we are in a room of 5 people, we can determine the number of handshakes by this line of reasoning: The first person will shake 4 hands (she won’t shake her own). The second person will shake 3 hands (he won’t shake his own of the hand of the first person, they already shook). The third person will shake 2 hands (same reasoning). The fourth person will shake 1 hand (that of the fifth person). The fifth person will shake 0 hands. So there will be a total of 1+2+3+4=10 handshakes

Answer 1

Missing part of the question

Determine the number of handshakes, i, that will occur for each number of people, n, in a particular room. (people)

Answer:

$S_n = \frac{n}{2}(n - 1)$

Step-by-step explanation:

Given

For 5 people

$\begin{array}{cc}{People} & {Handshakes} & {5} & {4} & {4} & {3} & {3} & {2} & {2} & {1} & {1} & {0} &{Total} & {10} \ \end{array}$

Using the given instance of 5 people, the number of handshakes can be represented as:

$(n - 1) + (n - 2) + (n - 3) + ........ + 3 + 2 + 1 + 0$

The above sequence is an arithmetic sequence and the total number of handshakes is the sum of n terms of the sequence.

$S_n = \frac{n}{2}{(T_1 + T_n})$

Where

$T_1 = n - 1$ --- The first term

$T_n = 0$ --- The last term

So:

$S_n = \frac{n}{2}(n - 1 + 0)$

$S_n = \frac{n}{2}(n - 1)$