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(17. Give an example of a problem whose solution uses the Principle of Inclusion-Exclusion and whose...
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 w...
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...
Problem 1. (4 pts) Combinatorics and the Principle of Inclusion Exclusion (a) (2pts) Roll a fair...
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....
Read the following problem. Use your knowledge about the Inclusion-Exclusion Principle to support your criteria. Telephone...
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,...
using the principle of inclusion-exclusion, find the number of solutions of the equation u1+u2+...+u6 = 15, where ui<6,i=1,..,6.
Give an example of a retirement planning problem whose solution involves calculating the present value of...
Give an example of a retirement planning problem whose solution involves calculating the present value of multiple cash flows. Give another example whose solution involves calculating the future value of multiple cash flows.
Problem 5, 10 points Roll three (6-sided) dice. Use inclusion-exclusion to find the probability that at...
Problem 5, 10 points Roll three (6-sided) dice. Use inclusion-exclusion to find the probability that at least one value of "2" appears. Hint: Consider A, to be the event that the ith dice shows a "2" for i 1,2,3. We want to find P(A1 UA2U A3) using PI.E. for 3 events. You can assume that each dice is fair, that is, P(A) 1/6, P(Ai n A) 1/6x 1/6-1/36 and P(An A2nA3) (1/6)3 1/216. For an easier solution, consider the complement...
The Pauli Exclusion Principle In this problem we will try to see how the requirement that...
The Pauli Exclusion Principle In this problem we will try to see how the requirement that the multi-electron wave function be antisymmetric under the exchange of two particles is related to the more familiar “Pauli exclusion principle”. Consider two non-interacting spin-particles in a 1D box of length 1, with the two particles in the excited configuration: V - 1012(x1) 02a(x1)| Y = € 1010(x2) 02a(2) | Take 01 = V2 sin(1x) and $2 = V2 sin(2tex). a) Verify that Ye...
Give an example of a spring system whose motion would be described by the solution to...
Give an example of a spring system whose motion would be described by the solution to the following initial value problem. Make sure to include units (you can choose whatever units you like, but they have to make sense and be consistent which each other). 21 x′′ + 12 x′ + 6x = 3 cos(2t) x(0) = −2 x′(0) = −1 (In other words: I am asking you to work backwards and give an example of a word problem that...
The Pauli Exclusion Principle tells us that no two electrons in an atom can have the...
The Pauli Exclusion Principle tells us that no two electrons in an atom can have the same four quantum numbers. For an electron in the 2s orbital shown above, enter a possible value for each quantum number n= m = Give ONE example. m, = Give ONE example. et visited Though a given electron only has one value for my, there are possible mvalues for electrons in 2s orbitals. Submit Answer Retry Entire Group 4 more group attempts remaining
Please do only Problem 4! Use 3 as result. 3. Use the inclusion-exclusion formula derived in...
Please do only Problem 4! Use 3 as result.
3. 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 j-1 This upper bound is referred to as the union bound. 4. Extend the above result to show that we have the analogous bound P( A) P(A), j-1 for the case of an arbitrary, but countable, number of events } Hint: Use the...
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....
Give an example of a retirement planning problem whose solution involves calculating the present value of multiple cash flows. Give another example whose solution involves calculating the future value of multiple cash flows.
Problem 5, 10 points Roll three (6-sided) dice. Use inclusion-exclusion to find the probability that at least one value of "2" appears. Hint: Consider A, to be the event that the ith dice shows a "2" for i 1,2,3. We want to find P(A1 UA2U A3) using PI.E. for 3 events. You can assume that each dice is fair, that is, P(A) 1/6, P(Ai n A) 1/6x 1/6-1/36 and P(An A2nA3) (1/6)3 1/216. For an easier solution, consider the complement...
The Pauli Exclusion Principle In this problem we will try to see how the requirement that the multi-electron wave function be antisymmetric under the exchange of two particles is related to the more familiar “Pauli exclusion principle”. Consider two non-interacting spin-particles in a 1D box of length 1, with the two particles in the excited configuration: V - 1012(x1) 02a(x1)| Y = € 1010(x2) 02a(2) | Take 01 = V2 sin(1x) and $2 = V2 sin(2tex). a) Verify that Ye...
The Pauli Exclusion Principle tells us that no two electrons in an atom can have the same four quantum numbers. For an electron in the 2s orbital shown above, enter a possible value for each quantum number n= m = Give ONE example. m, = Give ONE example. et visited Though a given electron only has one value for my, there are possible mvalues for electrons in 2s orbitals. Submit Answer Retry Entire Group 4 more group attempts remaining
Please do only Problem 4! Use 3 as result.
3. 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 j-1 This upper bound is referred to as the union bound. 4. Extend the above result to show that we have the analogous bound P( A) P(A), j-1 for the case of an arbitrary, but countable, number of events } Hint: Use the...
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