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

53 views5 pages
13 Aug 2016
Department
Course
Professor
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
Unlock document

This preview shows pages 1-2 of the document.
Unlock all 5 pages and 3 million more documents.

Already have an account? Log in
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
Unlock document

This preview shows pages 1-2 of the document.
Unlock all 5 pages and 3 million more documents.

Already have an account? Log in

Get access

Grade+
$10 USD/m
Billed $120 USD annually
Homework Help
Class Notes
Textbook Notes
40 Verified Answers
Study Guides
1 Booster Class
Class+
$8 USD/m
Billed $96 USD annually
Homework Help
Class Notes
Textbook Notes
30 Verified Answers
Study Guides
1 Booster Class