Binomial coefficients and combinatorial identities. Discrete Mathematics.
Answer question A) and question B)
Binomial coefficients and combinatorial identities. Discrete Mathematics. Answer question A) and question B) Exercise 1...
Discrete Math Question 1: Answer the following questions using your knowledge of binomial coefficients. Imagine a committee comprised of 7 men and 8 women. a) How many ways can you choose single representative from the committee? b) How many ways can you choose a task force of 3 members from the committee? c) How many ways can you choose a task force of 3 members who will then fit three roles: task force leader, task force vice-leader and task force...
Discrete mathematics question Can you please answe the following question? Please show your answer clearly. et n be a positive integer. Use the Master Theorem to obtain the big-O class for the functions that satisfy the following recurrences. (a) (4 points) g(n) -4g(n/2)+ n b) (4 points) (n) 2f (n/3) 0(n)
3. (12 points) Consider the following sum: n Sn = {(i + 1)(i +2) i=0 (a) Use properties of summations to find a closed form expression for Sn. Simplify your answer into a polynomial with rational coefficients. Show your work, and clearly indicate your final answer. (b) Use weak induction to prove that your closed form works for every integer n > 0. Make sure you include all three parts, and label them appropriately!
Discrete Mathematics Question 1: (a) Use the method of generalizing from the generic particular in a direct proof to show that the sum of any two odd integers is even. See the example on page 152 (4th edition, Discrete Mathematics with Applications) for how to lay this proof out. (b) Determine whether 0.151515... (repeating forever) is a rational number. Give reasoning. (c) Use proof by contradiction to show that for all integers n, 3n + 2 is not divisible by...
(A and C) Exercise 1.14. If n and k are integers, define the binomial coeffi- cient (m), read n choose k, by n! if 0 <k <n, = 0 otherwise. k!(n - k)! (a) Prove that ("#") = (m) + (-2) for all integers n and k. (b) By definition, () = 1 if k = 0 and 0 otherwise. The recursion relation in (a) gives a computational procedure, Pascal's triangle, for calculating binomial coefficients for small n. Start with...
Question 4 (a) Find the DFT of the series x[n)-(0.2,1,1,0.2), and sketch the magnitude of the resulting spectral components [10 marks] (b) For a discrete impulse response, h[n], that is symmetric about the origin, the spectral coefficients of the signal, H(k), can be obtained by use of the DFT He- H(k)- H-(N-1)/2 Conversely, if the spectral coefficients, H(k), are known (and are even and symmetrical about k-0), the original signal, h[n], can be reconstituted using the inverse DFT 1 (N-D/2...
2. Let f(x 11 k 1 k-0 (a) Give the interval of convergence (b) Find a closed form for f(x) on the interval of convergence. Theorem 35: The series Eanbn converges if (a) The partial sums An of Ean are bounded, (b) bob1b2 (c) lim,00 bn = 0 0, 7
Could I have help with entire question please. P+1 pt1 for any 2. In this question we will show by first principles that xpdz = p>0 a) Prove that (b) Use the formula (k +1)3- k3k23k +1 repeatedly to show that (for any n) m n (n+1) 7n and thus k2 mav be written in terms ofk- . Specifi- k-1 cally rL Note: An induction argument is not required here. (c) Using the same method with (complete) induction, or otherwise,...
6. In this question, you are going to study the approximation to binomial probabilities using the nor mal distribution. The binomial distribution is discrete while the normal distribution is continuous Therefore, we would expect some issues with approximating the binomial with the normal. (a) (2 points) Suppose X ~ Bin (25,04). Calculate E (N) and Var . (b) (4 points) Use the central lit theorem along with (a) to approximate Pr (X 8). Compare this with your result in #4(a)....
Main topic and problems for the final project The main purpose of the project is to introduce you how to use a in an computer as a research tool Introductory Discrete Mathematics. In this project you will be asked to show how the Fibonacci sequence {Fn} is related to Pascal's triangle using the following identities by hand for small and then by computers with large n. Finally, to rove the identity by mathematical arguments, such as inductions or combinatorics. I...