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There are 8 teams in a certain league and each team plays each of the

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There are 8 teams in a certain league and each team plays each of the [#permalink]

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New post 03 Dec 2009, 01:07
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There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played?

A. 15
B. 16
C. 28
D. 56
E. 64

OPEN DISCUSSION OF THIS QUESTION IS HERE: there-are-8-teams-in-a-certain-league-and-each-team-plays-134582.html
[Reveal] Spoiler: OA

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Last edited by Bunuel on 05 Feb 2015, 04:13, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: There are 8 teams in a certain league and each team plays each of the [#permalink]

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New post 03 Dec 2009, 07:35
its 8C2
that means
[(Factorial 8)] / [ (Factorial 2)(Factorial 6) ]
added OG tag :)
ro86 wrote:
There are 8 teams in a certain league and each team plays each of the other teams exactly once. if each game is played by 2 teams, what is the total number of games played?

1.15
2.16
3.28
4.56

OA is 28

I am not able to understand the explanation given to me in the OG.
The OG has calculated the answer as (8x7)/2.
has the og used some formula to arrive at the answer.

Tho i got the answer through manual counting. can any 1 explain the process used in the OG.

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Re: There are 8 teams in a certain league and each team plays each of the [#permalink]

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New post 03 Dec 2009, 07:49
ro86 wrote:
There are 8 teams in a certain league and each team plays each of the other teams exactly once. if each game is played by 2 teams, what is the total number of games played?

1.15
2.16
3.28
4.56

OA is 28

I am not able to understand the explanation given to me in the OG.
The OG has calculated the answer as (8x7)/2.
has the og used some formula to arrive at the answer.

Tho i got the answer through manual counting. can any 1 explain the process used in the OG.


this is nothing but application of direct formulae i.e combination ( if one is aware of this)
the formula is if we are asked to select r items out of n items then we can say nCr=n!/[(n-r)!*(n!)]
now in this q to play a match we need 2(r) teams out of 8(n) teams substituting
we get 8!/[2!*6!]=8*7*6!/[2!*6!]=8*7/2=28
HTH
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Re: There are 8 teams in a certain league and each team plays each of the [#permalink]

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New post 03 Dec 2009, 07:54
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This is just applying the idea of choosing combinations i.e. in how many ways can you choose 2 teams from a total 8. You can readily apply the formula for choosing 2 out of 8 as \(= (8x7)/2 = 28\)

The explanation would be something like:
Essentially, we need to find all possible pairs of teams from the 8.

So to create a pair, the first team can be chosen from the total of 8 - so 8 options/ways available. Now for each of this, we can choose the the opponent team for the pair from the 7 teams left. So the total will be \(8 x 7 = 56.\)

Now you can see that when we do the above, we will all possible pairs - (team1, team2), (team2, team3)....etc. We will also get the reverse pair (team2, team1), (team3, team2).... The order shouldn't matter since a game is just between 2 teams irrespective of the order.

So for every such pair of team, the reverse order is also included in the total of 56. So the total number of unique pairs we can have (which is the number of games) \(= 56/2 = 28\).
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Re: There are 8 teams in a certain league and each team plays each of the [#permalink]

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New post 03 Dec 2009, 12:02
Adding to the same question if each team plays twice with the other team then the answer would be 8*7=56 games.

Suppose the question states that a game is played by three teams, then will the total no of games played be (8*7*6)/(3*2*1).

Please correct me if i am wrong.
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Re: There are 8 teams in a certain league and each team plays each of the [#permalink]

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New post 24 Jul 2011, 02:03
I was so confused by this line "If each game is played by 2 teams, what is the total number of games played?

I got 28 (coz 7!) then when I see this line "If each game is played by 2 teams, what is the total number
of games played?" I took one extra step to divide 28/2 = 14......although 14 is not a choice but I wonder can ayone pls explain what does this line mean?
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Re: There are 8 teams in a certain league and each team plays each of the [#permalink]

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New post 25 Jul 2011, 10:05
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You can use combinations or you can do the following (note that this often comes up as something called "the handshake problem" as well as in counting the number of diagonals in convex polygons).

Team 1: plays 7 other teams - 7 games
Team 2: plays 7 other teams, but you already counted their game against team 1 - 6 games
Team 3: plays 7 other teams, but you already counted their games against teams 1 and 2 - 5 games
And so on

7 + 6 + 5 + 4 + 3 + 2 + 1 = 28

If there are x people in a room, then for everyone to shake hands with everyone else exactly once there will be (x(x-1))/2 handshakes. Same method with the polygon problem.
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Re: There are 8 teams in a certain league and each team plays each of the [#permalink]

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Re: There are 8 teams in a certain league and each team plays each of the [#permalink]

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New post 05 Feb 2015, 04:14
Expert's post
ro86 wrote:
There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played?

A. 15
B. 16
C. 28
D. 56
E. 64

OPEN DISCUSSION OF THIS QUESTION IS HERE: there-are-8-teams-in-a-certain-league-and-each-team-plays-134582.html


The total # of games played would be equal to the # of different pairs possible from 8 teams, which is \(C^2_{8}=28\).

Answer: C.
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Re: There are 8 teams in a certain league and each team plays each of the [#permalink]

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New post 05 Feb 2015, 04:14
Expert's post
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BOOKMARKED
Bunuel wrote:
ro86 wrote:
There are 8 teams in a certain league and each team plays each of the other teams exactly once. If each game is played by 2 teams, what is the total number of games played?

A. 15
B. 16
C. 28
D. 56
E. 64

OPEN DISCUSSION OF THIS QUESTION IS HERE: there-are-8-teams-in-a-certain-league-and-each-team-plays-134582.html


The total # of games played would be equal to the # of different pairs possible from 8 teams, which is \(C^2_{8}=28\).

Answer: C.


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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: There are 8 teams in a certain league and each team plays each of the   [#permalink] 05 Feb 2015, 04:14
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