There are 20 players in a table tennis competition. Each player must play exactly one match against every other player. How many matches are there in the competition?
number of games = 20(19)/2 = 190
We have to divide by 2 since games like A vs B and B vs A are the same game.
To determine the number of matches in the competition, we can use a combination formula.
Given that there are 20 players, each player needs to play against every other player.
Using the combination formula "n choose 2" (read as nC2), we can calculate the total number of matches as:
nC2 = n! / (2!(n-2)!)
Plugging in the value of n = 20:
20C2 = 20! / (2!(20-2)!)
Simplifying this expression:
20! / (2! * 18!)
Since 2! = 2 and 18! cancels out from 20!:
20 * 19 / 2
Therefore, the total number of matches in the competition is 190.
To find the number of matches in the table tennis competition, we need to calculate the number of combinations of 2 players that can be formed from a group of 20 players. This is because each match involves 2 players.
The formula to calculate the number of combinations of n objects taken r at a time is:
C(n, r) = n! / (r!(n - r)!)
In this case, we want to find C(20, 2). Let's calculate it step by step:
1. Calculate 20! (20 factorial). This means multiplying all the numbers from 1 to 20 together.
20! = 20 * 19 * 18 * ... * 3 * 2 * 1
2. Calculate 2! (2 factorial). This means multiplying all the numbers from 1 to 2 together.
2! = 2 * 1
3. Calculate (20 - 2)! [(n - r) factorial]. Subtracting 2 from 20 gives us 18.
(20 - 2)! = 18 * 17 * 16 * ... * 3 * 2 * 1
4. Substitute the values into the combination formula:
C(20, 2) = 20! / (2!(20 - 2)!)
= (20 * 19 * 18 * ... * 3 * 2 * 1) / ((2 * 1)(18 * 17 * 16 * ... * 3 * 2 * 1))
5. Simplify the expression:
C(20, 2) = (20 * 19) / 2
= 190
Therefore, there are 190 matches in the table tennis competition with 20 players.