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.