Value of x

Question

There are x number of teams in a certain league. Each team plays against the other teams only once. 2 teams play one match and the total number of matches are 28. what is the value of x? 

Answer 1

The combination of 2 from X will equal the total matches played. 

 

so [x!/(x-2)!2!]=28

since X! = X (X-1)(X-2)!

the (X-2)! Will cancel and the numerator becomes x(X-1) and 2! Is just 2 so simplified we get

x(X-1)=56

X² -X -56=0

(X-8)(X+7)=0

x=8,-7 but since the matches must be >0 only 8 is valid. 

 

So so there are 8 teams and each team will play 7 games and divide it by 2 bc each game has 2 teams so 28 matches. 

Answer 2

First, you have to decide whether you are talking about permutations or combinations. Does the order matter? Is it the same or different if you have Team A v Team B or Team B v Team A?

 

Team A v Team B is the same match up as Team B v Team A, since each teams plays another only once. The order doesn't matter, so you are using a combination.

 

The formula for combinations is _nC_r=n!/[(n-r)!r!] where n represents the number of items being chosen from (in this case the number of teams) and r represents how many are chosen at a time (in this case, 2). You know that the result is 28, so we can set up an equation like this:

_nC₂=28

     n!      =28

(n-2)!2!

 

Since 2!=2*1=2, we can multiply both sides of the equation by 2 to result in:

   n!      =56

(n-2)!

 

Let's take a look at some real number examples to see what this means. Let's let n equal 4, 5, and 6.

If n=4, 4! = 4*3*2*1   = 4*3  =12

            2!           2*1  

If n=5, 5!   =  5*4*3*2*1   =5*4 = 20

           3!               3*2*1

If n=6, 6! = 6*5*4*3*2*1 =6*5 =30

           4!            4*3*2*1

 

As you can see, this pattern indicates that n!    will equal n*(n-1).

                                                            (n-2)!

We can then simplify our combination equation to read n(n-1)=56. You can quickly guess and check in your mind at this point to come up with the answer, or you can continue the algebra:

n²-n=56

n²-n-56=0

(n-8)(n+7)=0

n=8 or -7, but since you can't have negative numbers of teams, n=8.

 

Check this into the original combination, and you can see that there are indeed 8 teams in the league.