Balls and Urns
“Balls and urns” problems are paradigmatic. Many prob
lems can be recast as balls and urns problems, once we
fgure out which are the balls and which are the urns.
How many ways are there oF putting
b
balls into
u
urns?
•
That depends whether the balls are distinguishable
and whether the urns are distinguishable
How many ways are there oF putting 5 balls into 2 urns?
•
IF both balls and urns are distinguishable: 2
5
= 32
◦
Choose the subset oF balls that goes into the frst
urn
◦
Alternatively, For each ball, decide which urn it
goes in
◦
This assumes that it’s OK to have 0 balls in an
urn.
•
IF urns are distinguishable but balls aren’t: 6
◦
Decide how many balls go into the frst urn: 0, 1,
. . . , 5
•
IF balls are distinguishable but urns aren’t: 2
5
/
2 = 16
•
IF balls and urns are indistinguishable: 6
/
2 = 3
1
What iF we had 6 balls and 2 urns?
•
IF balls and urns are distinguishable: 2
6
•
IF urns are distinguishable and balls aren’t: 7
•
IF balls are distinguishable but urns aren’t:
2
6
/
2 = 2
5
•
IF balls and urns are indistinguishable: 4
◦
It can’t be 7
/
2, since that’s not an integer
◦
The problem is that iF there are 3 balls in each
urn, and you switch urns, then you get the same
solution
2
Reducing Problems to Balls and Urns
Q1:
How many di±erent confgurations are there in Tow
ers oF Hanoi with
n
rings?
A:
The urns are the poles, the balls are the rings. Both
are distinguishable.
Q2:
How many solutions are there to the equation
x
+
y
+
z
= 65, iF
x, y, z
are nonnegative integers?
A:
You have 65 indistinguishable balls, and want to put
them into 3 distinguishable urns (
x
,
y
,
z
).
Each way oF
doing so corresponds to one solution.
Q3:
How many ways can 8 electrons be assigned to 4
energy states?
A:
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 Spring '07
 SELMAN
 Product Rule, Probability, Electron, How Many Ways, urn

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