# MACM 101 Lecture Notes - Lecture 6: Anagram, Binomial Theorem

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13 Aug 2016
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In order to join her drill team, Deborah must shoot 12 of 12 clay targets using all 12 bullets in
her pistol, 1 bullet per target. There are 4 red, 3 white, 2 green, and 3 blue targets as shown in
the figure.
In how many different orders can Deborah shoot the targets?
(must shoot bottom-most target first for any colour, otherwise targets below it will be lost)
4 x 4 x ? …
Strategy: use a 12-letter anagram
Anagram has: 4 x R, 3 x W, 2 x G, 3 x B
The first instance of each letter means that she shoots the bottom-most remaining target in that column
Bananas in Lunchboxes
E.g. There are 10 identical bananas to be distributed among 5 different lunchboxes. How many different
ways can this be done?
There are 5 boxes, so line up the 10 fruits …
… and insert 4 dividers amongst them!
10 undistinguishable bananas, 4 undistinguishable dividers; order is unimportant
Lecture 6
January 18, 2016
4:56 PM
Lecture Notes Page 1
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Variant: suppose each lunchbox must contain at least one fruit. Now, how many ways are there?
Solution:
Nail down 1 fruit in each box. Now distribute the remaining 5.
Arrange: 5 fruits, 4 dividers =>
Variant 2: what if some of the 10 fruits could be left over?
Solution: add extra box/divider for left over bananas
Arrange: 10 fruits, 5 dividers =>
Equivalent Problem: How many nonnegative integer solutions are there to the following equation?
Solution: think of it as 40 bananas in 3 lunchboxes or 40 "objects" and 2 "dividers" =>
Binomial Theorem
The coefficients in the binomial expansion (x + y)nare exactly
Lecture Notes Page 2
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