Mathematics 1228A/B Lecture 1: Section 1.5 Sep 23, 2019

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Published on 2 Aug 2020
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1.5 Combinations: subsets of a given Size!
!
"A subset of a specified size is called a combination.!
"eg. Combination of 2/5 objects means subset with (only exactly) 2 of the objects. !
!
#of permutations of k out of n objects!
= # of ways to choose k of the n objects * #of permutations of the k chosen objects!
So n!/(n-k)!=# of ways to choose k of n objects *k!!
!
#of ways to choose k out of n objects =1/k!*n!/(n-k)!= n!/k!(n-k)!!
!
!
For any non-negative integers n and k, with k <=n, !
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The number of dierent ways of choosing a subset of k objects from of set of n distinct objects is in fact !
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Example 1.15!
The Drama Club is holding a lottery to raise money. They will draw 4 winning lottery tickets, and the owner of each winning
ticket will receive a pair of vouchers which can be traded in for tickets to future dramatic performance by the Drama Club. If 12
lottery tickets have been sold, in how many ways can the 4 winning tickets be selected.!
!
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Properties of !
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Example 1.16!
The Drama Club in the previous example is actually 8 year old suzie and her little brother Jimmy. Suzie made 12 tickets and
then harassed her parents until between them they had bought all the tickets for 10 cents each. Mom bought 8 tickets and dad
bought the other 4!
1) in how many ways could the 4 winning tickets be selected so that each parents wins 2 prizes.!
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2) in how many ways could the winning tickets be selected so that mom wins at least 3 prizes!
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Document Summary

A subset of a speci ed size is called a combination. eg. combination of 2/5 objects means subset with (only exactly) 2 of the objects. #of permutations of k out of n objects. = # of ways to choose k of the n objects * #of permutations of the k chosen objects. So n!/(n-k)!=# of ways to choose k of n objects *k! #of ways to choose k out of n objects =1/k! For any non-negative integers n and k, with k <=n, The number of di erent ways of choosing a subset of k objects from of set of n distinct objects is in fact. The drama club is holding a lottery to raise money. They will draw 4 winning lottery tickets, and the owner of each winning ticket will receive a pair of vouchers which can be traded in for tickets to future dramatic performance by the drama club.

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