Math question: how many matches?

Question

There are 14 tennis players, all of which must play one match against the other players. I know the answer is 104 by using the the following math:

14+13+12+11+10+9+8+7+6+5+4+3+2.

But there HAS to be a more intelligent way to calculate this right? I can't remember it, and my working backwards isn't helping! Thanks :)

Answer 1

= ([14 - 1]² + [14 - 1])/2

= (13² + 13)/2

= (169 + 13)/2

= 182/2 or 91

Answer: 91 is the total number of matches.

Proof:

= 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

= 91

Why 13 is the highest number considered? Because no player can play against his own self—no player matches himself versus his very own self.

Answer 2

Use Permutations....

Take note that you may pair them. However, there would be chances were you will be pairing the same players twice..

P(14,2)/2 = (14!/12!)/2 = 91... ANS

Answer 3

Not 14 but need 1 more.

Yes, the answer is 91.

By pairing 13 and 1, 12 and 2, etc...

so it is  14 *13 /2 = 91

Answer 4

I believe you are seeking the answer to the following question:

How many combinations (C) are there of 14 players (p) taken 2 at a time (r).

The answer is: C = C(p,r) = p!/[r!*(p - r)!]

So, C = 14!/[2!*12!] = 7*13 = 91 (not 104)

Note: Try it out, say for p = 4. The combos are 12,13,14,23,24,34.

For this case, the number of combos is obviously 6. The C formula for this case gives 4!/[2!*(4 - 2)!] = (4*3)/2! = 6, which demonstrates the C formula works for p = 4, r = 2. You can verify the formula works for any other case(s) of further interest.