Question

in a soccer season, each team must play every other team exactly once. complete the table that shows the number of teams in the tournament and the number of games played.

number of teams	2	3	4	5	6	...	10
number of games						...

Explanation:

Step1: Use combination formula

The number of games is the number of ways to choose 2 teams out of $n$ teams for a match. The combination formula is $C(n,k)=\frac{n!}{k!(n - k)!}$, where $n$ is the total number of items and $k$ is the number of items to choose. Here $n$ is the number of teams and $k = 2$. So the number of games $G=\frac{n(n - 1)}{2}$.

Step2: Calculate for $n = 2$

When $n=2$, $G=\frac{2\times(2 - 1)}{2}=\frac{2\times1}{2}=1$.

Step3: Calculate for $n = 3$

When $n = 3$, $G=\frac{3\times(3 - 1)}{2}=\frac{3\times2}{2}=3$.

Step4: Calculate for $n = 4$

When $n = 4$, $G=\frac{4\times(4 - 1)}{2}=\frac{4\times3}{2}=6$.

Step5: Calculate for $n = 5$

When $n = 5$, $G=\frac{5\times(5 - 1)}{2}=\frac{5\times4}{2}=10$.

Step6: Calculate for $n = 6$

When $n = 6$, $G=\frac{6\times(6 - 1)}{2}=\frac{6\times5}{2}=15$.

Step7: Calculate for $n = 10$

When $n = 10$, $G=\frac{10\times(10 - 1)}{2}=\frac{10\times9}{2}=45$.

Answer:

Number of teams	2	3	4	5	6	...	10