SOLUTION: Eight teams are to play against each other five times in a tournament how many matches will be played?

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Question 1031173: Eight teams are to play against each other five times in a tournament how many matches will be played?
Answer by Bakr.R(8) About Me  (Show Source):
You can put this solution on YOUR website!
This question is extremely simple, though you need to just realize the situation of the problem, just focus and realize, then it will be easy to figure out, for now, and next time a similar question comes up.

Eight teams are to play against EACH OTHER, five times, and so what are the total of matches played?

First, to start simple, let us assess one team, and this will be our 1st team, let's call it Team 1. Team 1 is to play against each team five times, we have eight teams, but Team 1 is only playing against SEVEN other teams (since Team 1 is part of the eight teams), and so for Team 1, it is playing against all other seven teams 5 times.
Team 1:7%2A5=35

Now, let's move to Team 2. Team 2 is playing against all other seven teams (again, because its part total eight teams), five times, but, it already played 5 matches with Team 1 already, right? because when we calculated the total of matches played by Team 1, we already counted in the matches it played with Team 2, five matches, so, that means, we cannot repeat it here. So, we remove one team from the total.
Team 2:6%2A5=30 6, because it only played 5 matches against the six other teams, and we've already counted in the match it played with Team 1 in the previous calculation, so by making it a 7, we're saying that Team 1 and Team 2 played a total of 10 matches, and that is incorrect.

Now, Let's move on to Team 3. It is must play 5 matches against each team, but, we have already counted its matches BOTH with Team 1 AND Team 2 (since in the calculations of Team 1 and team 2, we calculated the match it played with ALL other teams, and Team 3 is one of them), so, we only calculate the numeber of matches with the rest of the teams it hasn't played with, which are a total of 5 teams. Do you see the pattern here?
Team 3:5%2A5=25 Again, we did this since we already calculated the matches Team 3 played with both Team 1 and Team 2, so we do not count it again.

And so, if you've realized the pattern, and realized the situation of the problem, it will make sense to you.

Now, we may deduce an expression for each calculation, when assessing each team,
Team 1:7%2A5=35 Team 5:3%2A5=15
Team 2:6%2A5=30 Team 6:2%2A5=10
Team 3:5%2A5=25 Team 7:1%2A5=5
Team 4: 4%2A5=20

Now, why did we stop at Team 7? shouldn't we still have a calculation for Team 8? Actually, that would be incorrect, since this is like saying that Team 8 is playing 5 matches with..... no one, a non existent team, but should we still have a calculation for Team 8? Well, actually, we already calculated all of its matches played with all other teams; since we already calculated for each team, the matches it played with all other teams, and Team 8 was included in each calculation, so, calculating any matches now with Team 8, would repeat matches that we've already counted, so that would be incorrect. and yet the calculation would be Team 8: 0%2A5=0 0 matches
So now, with all the calculations, we reach to an expression:

So, the total number of matches played is 140.

In any probability question or problem, always think how to understand the situation, usually it requires more of a logical approach than an algebraic method or equation, to solve, keep this in mind, and it will become much easier.
Good luck!