Use the inclusion – exclusion principle to find the number of primes less than 24. Use...
Read the following problem. Use your knowledge about the Inclusion-Exclusion Principle to support your criteria. Telephone numbering is an application of the inclusion-exclusion principle. Discuss with your peers a way in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry). For each new numbering plan, you discuss show how to find the number of different telephone numbers...
using the principle of inclusion-exclusion, find the number of solutions of the equation u1+u2+...+u6 = 15, where ui<6,i=1,..,6.
Read the following problem. Use your knowledge about the Inclusion-Exclusion Principle to support your criteria. Telephone numbering is an application of the inclusion-exclusion principle. Discuss with your peers a way in which the current telephone numbering plan can be extended to accommodate the rapid demand for more telephone numbers. (See if you can find some of the proposals coming from the telecommunications industry). For each new numbering plan you discuss show how to find the number of different telephone numbers...
Exercise 4. By writing AU BUC as (AUB) UC, show that the Principle of Inclusion-Exclusion for three sets is P(AUBUC) = P(A)+P(B)+P(C)- P(ANB) - P(ANC) - P(BNC)+P(ANBNC) Can you generalize the result to an arbitrary number of events?
(17. Give an example of a problem whose solution uses the Principle of Inclusion-Exclusion and whose answer is 37 - 34 +1 (= 3.36 – 3.33 + 1).
Use the inclusion-exclusion formula derived in class as well as induction on the integer n to show that for any sequence of events {Ajli, we have that P( A) IPA). j-1 This upper bound is referred to as the union bound.
Problem 1. (4 pts) Combinatorics and the Principle of Inclusion Exclusion (a) (2pts) Roll a fair die 10 times. Call a number in 1, 2, 3, 4, 5, 6 a loner if it is rolled exactly once on the 10 rolls. (For example, if the rolls are 1 2 6 4 4 4 6 3 4 1, then 2 and 3 are the only loners) Compute the probability that at least one of numbers 1, 2, 3 is a loner....
2. Prove the three-set version of the inclusion-exclusion principle: using P(AUB)-P(A) + P(B)
5. Find all twin primes less than 100, and find π(100).
8. Define (n) to be the number of positive integers less than n and n. That is, (n) = {x e Z; 1 < x< n and gcd(x, n) = 1}|. Notice that U (n) |= ¢(n). For example U( 10) = {1, 3,7, 9} and therefore (10)= 4. It is well known that (n) is multiplicative. That is, if m, n are (mn) (m)¢(n). In general, (p") p" -p Also it's well known that there are relatively prime, then...