How many individual players are there altogether?

Question

At a high school there are 3 athletic teams. 21 on the basketball team, 26 on the baseball team and 29 on the soccer team. These players are not all different, in fact, 14 play both baseball and basketball, 15 play both baseball and soccer, 12 play soccer and basketball and 8 are on all three teams. How many individual players are there altogether?

I'm really confused!! Please explain thoroughly!

Oops! I meant 21, 26, 29 are the number of players on the teams!

I still don't get it...

Ohhh, now I get it, thanks Puzzling!

Answer 1

It's easiest to draw a Venn diagram (3 intersecting circles). Label them as Basketball, Baseball and Soccer.

There are 8 players that play all 3 sports, so put '8' in the middle.

From the totals for each pair of sports, subtract off 8 and put those in the intersection of two circles. So for the part of the intersection between baseball and basketball, put 6 (since 8 are already accounted for). Similarly for baseball and soccer, put 7 in between those two circles. And finally for soccer and basketball, put 4.

Finally, put the remaining players that only play on one team in the circles where they don't overlap. So for basketball you have 21 players, but you have already listed 18 as playing in 2 or 3 sports. That leaves 3 players that *only* play basketball. You can do the same thing with baseball (5) and soccer (10).

The diagram will look like the attached picture. Add up all the individual numbers to get your answer. If I've got your information correctly, the answer is 43 individual players.

Another way to get this is to add the total players for the 3 teams (21 + 26 + 29 = 76). But this overcounts the players that play on multiple teams, so from this subtract off the numbers for players on 2 teams (14 + 15 + 12 = 41). That leaves 35. But now you have undercounted because there are some players on all 3 teams. So add 8 back in. Ah! The answer is 43 either way!

Answer 2

Group A=Basketball Team
Group B=Baseball Team
Group C=Soccer Team
n(A)=21
n(B)=26
n(C)=29
n(A and B in common=D)=14
n(B and C in common=E)=15
n(A and C in common=F)=12
n(A and B and C in common=G)=8
-Now how to solve:
all of the individuals=n(A)+n(B)+n(C)-n(D)-n(E)-n(F)+n(G)
=21+26+29-14-15-12+8
=43

Answer 3

Following Puzzling, I came up with 43. Unfortunately, I cannot draw a diagram.