# 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?

Solution:

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

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