# Organized Counting with Venn Diagrams

• Nov 23rd 2009, 11:50 AM
john-1
Organized Counting with Venn Diagrams
The Vennville junior hockey team has 12 members who can play forward, 8 who can play defence, and 2 who can be goalies. What is the smallest possible size of the team if

a) no one plays both defence and forward?
b) three of the players are able to play defence or forward?
c) both the goalies can play forward but not defence?

i don't understand how the answers in the back of my text are derived. a) 20 b) 19 c)20
• Nov 24th 2009, 09:15 AM
Hello john-1
Quote:

Originally Posted by john-1
The Vennville junior hockey team has 12 members who can play forward, 8 who can play defence, and 2 who can be goalies. What is the smallest possible size of the team if

a) no one plays both defence and forward?
b) three of the players are able to play defence or forward?
c) both the goalies can play forward but not defence?

i don't understand how the answers in the back of my text are derived. a) 20 b) 19 c)20

a) No-one plays both defence and forward. So that means that none of the 12 who can play forward can play defence. So these 12 players are different people from the 8 who play defence. That means we must have at least 20 different people in the team.

Nothing is said about the goalies. So, provided 2 of the 20 that we already have counted can also play as goalies, we simply have these 20 as our minimum size team.

b) Of the twelve who can play forward, 3 can play defence. So there are another 5 who play defence but not forward. That's 12+5=17 players so far.

Again, nothing has been said about the goalies. So provided two of these 17 can play as goalies, 17 is the minimum size of the team. (I therefore disagree with the answer of 19 that you have been given.)

c) The 2 goalies can play forward but not defence. So the 12 who can play forward are all we need to account for the forwards and the goalies. Provided, then, that 8 of the 10 forwards who can't play in goal can play in defence, 12 is all we need to meet the criteria. (Again, that differs from the answer in the back of the book.)