datonman 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
As noted above, each game must be played by a UNIQUE PAIR of teams.
An alternate approach is to WRITE OUT all of the possible pairings.
Let the 8 teams be A, B, C, D, E, F, G and H.
Pairs with A:
AB, AC, AD, AE, AF, AG, AH.
7 options.
Pairs with B (excluding AB, which has already been counted):
BC, BD, BE, BF, BG, BH.
6 options.
Pairs with C (excluding AC and BC, which have already been counted):
CD, CE, CF, CG, CH.
5 options.
Notice the PATTERN.
Options for A = 7.
Options for B = 6.
Options for C = 5.
With each successive letter, the number of options decreases by 1.
Thus:
Options for D = 4.
Options for E = 3.
Options for F = 2.
Options for G = 1.
Adding together the options above, we get:
Total number of games = 7+6+5+4+3+2+1 = 28.
The correct answer is
C.
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