How many teams participated in the tournament?

Question

In a basketball tournament, 153 total matches are played. Each team played exactly one match with each of the other teams. How many teams participated in the tournament?

Answer 1

This question is not about Basketball it’s about permutations and combinations and that makes it a question for the math department. Games will not be one and lost because a player does or does not know how many teams are participating in the tournament. You might try to departmentalize your questions a little better.

Answer 2

Total # of combinations is xC2 or "x Choose 2" which must equal 153. --->

x! / [(x-2)! * 2!)] = 153

Multiply both sides by 2! --->

x! / (x-2)! = 306

Keep in mind, that x! is x * (x-1) * (x-2) * (x-3) * (x-y)....etc. until (x-y) = 1

So in the case of x! / (x-2)! , all of the numbers after (x-1) will cancel out (they are being divided) and leave us with --->

x (x-1) = 306

From here just use algebra to create a quadratic polynomial

x^2 - x - 306 = 0

From here, factor (find 2 numbers with a product of 306 and a sum of -1) --->

(x-18)(x+17) = 0

Solving for the zeros give us +18 and -17

-17 is obviously an answer that doesn't make logical sense in this case so the answer must be 18 teams.

Answer 3

This is a combination problem involving the number of matches played selecting 2 teams at a time from a given number of teams. 

Let n represent the number of teams. Therefore, using combination, we can write the Combination formula as follows:

 

nCr = n!/[(n- r)!r!] = n!/[(n-2)!2!]

 

This should be equal to 153 (the total number of matches played).

 

Simplifying the above yields:

n(n-1)/2 = 153

n² - n = 306

n² - n - 306= 0

Factor the left hand side and rewrite the equation:

(n -18)(n + 17) = 0

n-18 = 0 or n+17 = 0

 

The solution is 18. Choose only the positive answer.

 

The number of teams that participated in the tournament is 18.

 

Explanatory Notes

Since a team cannot play with itself, it will have to play with n-1 teams. Because each team can only play exactly one match, we have to divide by 2 to avoid duplication. Thus, in a problem like this, the expression for the combination is: n(n - 1)/2