Download presentation

Presentation is loading. Please wait.

Published byMax Ask Modified over 2 years ago

1
7.6 Applications of Inclusion/Exclusion Problem: Find the number of elements in a set that have none of n properties P1, P2,.. Pn.

2
Intro Ex: not divisible by 5 or 7 Div by 5Div by 7

3
Let A i =subset containing elements with property P i N(P 1 P 2 P 3 …P n )=|A 1 ∩A 2 ∩…∩A n | N(P 1 ’ P 2 ‘ P 3 ‘…P n ‘)= number of elements with none of the properties P1, P2, …Pn =N - |A1 A2 … An| =N- (∑|Ai| - ∑|Ai ∩ Aj| + … +(-1) n+1 |A1∩ A2 ∩…∩ An|) = N - ∑ N (Pi) + ∑(PiPj) -∑N(PiPjPk) +… +(-1) n N(P1P2…Pn)

4
N(P 1 ’ P 2 ‘ P 3 ‘…P n ‘) Number of elements with none of the properties P1, P2, …Pn N(P 1 ’ P 2 ‘ P 3 ‘…P n ‘) = N - ∑ N (Pi) + ∑(PiPj) -∑N(PiPjPk) +… +(-1) n N(P1P2…Pn)

5
Ex 1: How many solutions does x 1 +x 2 +x 3 = 11 have where xi is a nonnegative integer with x 1 ≤ 3, x 2 ≤ 4, x 3 ≤ 6 (note: harder than previous problems) Let P1: x1≥4, P2: x2≥5, P3: x3≥7 N(P 1 ’ P 2 ‘ P 3 ‘ ) = N – N(P1) – N(P2) – N(P3) + N(P1P2)+ N(P1P3)+N(P2P3) –N(P1P2P3) =13C11 – 9C7 – 8C6 – 6C4 + 4C2 + 2C0 + 0 – 0 =78 – 36 – 28 – 15 + 6 + 1 + 0 – 0 = 6

6
Check the solution Check: 3 4 4 3 3 5 1 4 6 2 3 6 2 2 6 3 2 6

7
Ex: 2: How many onto functions are there from a set A of 7 elements to a set B of 3 elements Let A={a1,a2,a3,a4,a5,a6,a7}, B={b1,b2,b3} Let Pi be the property that bi is not an element of range(f) N(P 1 ’ P 2 ‘ P 3 ‘ ) = N – N(P1) – N(P2) – N(P3) + N(P1P2)+ N(P1P3)+N(P2P3) –N(P1P2P3)

8
answer =3 7 -3C1*2 7 +3C2*1 7 -0 =2187-384+3 =1806 Note: Theorem 1 covers these problems in general.

9
Application: How many ways are there to assign 7 different jobs to 3 different employees if every employee is assigned at least one job?

10
Sieve of Eratosthenes 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100

11
Sieve- on numbers 2 to 100 The number of primes from 2-100 will be 4+N(P 1 ’ P 2 ‘ P 3 ‘P 4 ‘ ) Define: P1: numbers divisible by 2 … P2: P3: P4: N(P 1 ’ P 2 ‘ P 3 ‘P 4 ‘ ) = N – N(P1) – N(P2) – N(P3) - N(P4) +N(P1P2)+ N(P1P3)+N(P1P4)+N(P2P3)+N(P2P3)+N(P3P4) –N(P1P2P3)-N(P1P2P4)-N(P1P3P4)-N(P2P3P4) +N(P1P2P3P4) =… =

12
Hatcheck problem Question: A new employee checks the hats of n people and hands them back randomly. What is the probability that no one gets the right hat. (Similar question: If I don’t know anyone’s names, what’s the probability no one gets the right test back?) Term: “Derangement”- permutation of objects that leave NO object in its original place

13
Examples of derangements For 1,2,3,4,5, make some examples of a derangement: Some non-examples:

14
Thm. 2: The number of derangements of a set with n elements is… Dn= n![1 - ] In order to prove this, we will use the N(P 1 ’ P 2 ‘ P 3 ‘…P n ‘) formula. But, first we calculate: N=___ N(Pi)=___ N(PiPj)=___

15
Proof of derangement formula Since N=n!, N(Pi)=(n-1)!, N(PiPj)=(n-2)!,… Dn = N(P 1 ’ P 2 ‘ P 3 ‘…P n ‘)= = N - ∑N(Pi) + ∑N(PiPj) + -∑N(PiPjPk) +…+(-1) n N(P1P2…Pn) =

16
Dn = N - ∑N(Pi) + ∑N(PiPj) + -∑N(PiPjPk) +…+(-1) n N(P1P2…Pn) =n! – nC1* (n-1)! + nC2*(n-2)!- … + (-1) n *nCn(n- n)! = … = n! – n!/1! + n!/2! - …+(-1) n n!/n! = n![1 - ]

17
Probability no one gets the same hat back… is Dn/n!= 1 – 1/1! + ½! - … (-1) n 1/n! As n , Dn/n! 1/e =.368… N2345 Dn/n!.5.33333.335.36666

18
Other applications Matching cards Mailboxes Passing tests back

Similar presentations

Presentation is loading. Please wait....

OK

Mathematical Maxims and Minims, 1988

Mathematical Maxims and Minims, 1988

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on obesity diet plan Ppt on different types of dance forms in africa Ppt on types of ram and rom Ppt on leadership development training Ppt on masculine and feminine gender for grade 3 Ppt on water pollution problems and solutions Ppt on indian politics democracy Ppt on different types of computer softwares Ppt on heterotrophic mode of nutrition in plants Ppt on bluetooth communication devices