Chapter 1.1, Problem 14STP

To determine

How many games does the president schedule.

Explanation of Solution

GIVEN:

Five new students join the chess club. The club president schedules games so that each of the new member plays a game against each of the other new members.

Concept used:

Combination formula ${}^{\text{n}}{\text{C}}_{\text{r}}=\frac{\text{n!}}{\text{r!}\left(\text{n}-\text{r}\right)\text{!}}$

In order to find how many games does the president schedule, using combination formula because the order of the two people chosen for each game not matter so it is a combination, there are $n=5$ people to choose from for each game and $r=2$ people are chosen, then the total number of games is:

  $\begin{array}{l}{}^{\text{n}}{\text{C}}_{\text{r}}=\frac{\text{n!}}{\text{r!}\left(\text{n}-\text{r}\right)\text{!}}\\ {}^5{\text{C}}_2=\frac{\text{5!}}{\text{2!}\left(5-2\right)\text{!}}\\ =\frac{5\text{!}}{2\text{!3!}}\\ =\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{2\cdot 1\cdot 3\cdot 2\cdot 1}\\ =\frac{120}{12}\\ =10\end{array}$

So, there are 10 number of games scheduled by the presdent.