Mathematics 1228A/B Lecture 1: Section 1.3 Sep 18, 2019

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Published on 2 Aug 2020
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The Fundamental Counting Principle!
!
Theorem:!
"If there are k decision to be made, with Mi choices available for the ith decision, for i =1...k!
No matter which choice is selected at some earlier decision, then the number of dierent ways of making all k decisions is m1
x m2 x ....mk!
!
!
Suppose there’s !
3 dierent burger and !
4 kinds of soft drink and !
3 choices of side orders and !
3 dierent sizes!
!
The # of ways to decide is 3x4x2x3=72 ways!
!
!
Example 1.6!
How many 3 digit codes can be formed from the digits 0-9!
"i) if repetition of digits within the code is allowed?!
!
!
"ii) if repetition is not allowed!
!
!
!
How many 3 digits codes can be formed if repetition is not allowed and the code must either start with an odd digit or else
must contain only even digits.!
!
!
!
!
!
Example 1.7!
"How many 3 digit codes have at least one repeated digits?!
!
!
!
!
!
!
Example 1.8!
!
1) How many subset of {a,b} are there?!
!
!
2) How many subset of {a,b,c} are there!
!
!
!
!
!
3) How many subset of (a, b, c, d, e, f) are there!
!
!
4) How many subset of {a, b,c,d,e,f} are there which contain the d but have no vowels.!
!
!
!
!
Theorem:!
A set containing n element has 2^n subsets
10 10 10 1000
10 98720
59872 5360
54360
1420710
1010 1000 possible
no rep 10 98e720
at least Irepeat tooo 220 280
0alb lab
0AibCab Ac bc iAbc
2Choicesfora2choicesfor b2choicesfo cIn Out
23
26 2222x22
dxz
abC
11 I
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Document Summary

No matter which choice is selected at some earlier decision, then the number of di erent ways of making all k decisions is m1 x m2 x mk. If there are k decision to be made, with mi choices available for the ith decision, for i =1k. The # of ways to decide is 3x4x2x3=72 ways. How many 3 digit codes can be formed from the digits 0-9: if repetition of digits within the code is allowed, if repetition is not allowed. How many 3 digits codes can be formed if repetition is not allowed and the code must either start with an odd digit or else must contain only even digits. 1010 1000 possible no rep 10 9 8 e 720 at least i repeat tooo 220 280. Example 1. 8: how many subset of {a,b} are there, how many subset of {a,b,c} are there. 0 a i b c ab ac bc i abc.

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