We have a game called "Mother Sheep" in which each player receives a
card with the names of 5 sheep out of 10 sheep in the game and has to
build fences to enclose those 5. After playing, Perry asked whether,
if every card were dealt out and 5 sheep were enclosed, then some
player would have won. This led to a conversation about how many ways
there are to choose 5 sheep out of 10. At firs Perry tried to do it by
making patterns among the 10 sheep shown on a card, but that did not
work. So I asked him if he wanted some guidance, and the following
conversation ensued:
Ken: How many ways are there to choose one sheep?
Perry: 10.
K: How many ways are there to choose 2 sheep if one of them is Bob?
P (counts the remaining sheep): 9.
K: How many ways are there to choose 2 sheep if one of them is
Tabitha?
P: 9.
K: So how many ways in all are there to choose 2 sheep?
P: 90.
K: Have we forgotten any possible pairs of sheep?
P: No.
K: How many times have we counted Bob and Tabitha?
P: 2
K: How many times have we counted each pair of sheep?
P: 2
K: So how many different ways are there to choose 2 sheep?
P: 55!
K: 90 divided by 2 is 45.
P: 45.
K: Now suppose we chose Bob and Tabitha. How many ways are there to
choose 3 sheep if 2 are Bob and Tabitha?
P (counts remaining sheep): 8
K: So how many ways are there to choose 3 sheep?
P: I need help. What's 45 times 8?
Judy: 360.
P: 360 ways.
K: How many times have we counted Bob and Tabitha and Norman?
P (count the 3 sheep): 3
K: So we should divide 360 by 3 to get 120 different ways of choosing
3 sheep. Suppose we have 3 sheep already. How many ways are there to
choose the fourth sheep?
P (counts): 7.
K: And how many times have we counted each group of 4?
P (who has got the hang of this now): 4
K: So let's divide 120 by 4 first to get 30, then multiply by 7 to get
210 ways of choosing 4 sheep. Now if we have 4 sheep, how many ways
can we choose the fifth?
P: 6
K: And how many times did we count each group of 5?
P: 5
K: So let's divide 210 by 5 first to get 42, then multiply by 6 to
get 252.
P: 252 ways of choosing 5 sheep! Even if you have all the cards
[there are only about 30], you still might not win.
Socrates would say that he knew it all along.