Questions 5,6,7,8 please. Thank you. Argue that the binomial coefficient (n n/2)

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Questions 5,6,7,8 please. Thank you.

Argue that the binomial coefficient (n n/2) is an even integer. Let m, n, k, and l be integers such that m greaterthanorequalto 1, n greaterthanorequalto 1, and 1 lessthanorequalto l lessthanorequalto k lessthanorequalto n. After a week of hard work. Elisa Kazan1 goes to her neighborhood pub. This pub has m different types of beer and n different types of cider on tap. Elisa decides to order k pints: At most one pint of each type, and exactly l pints of cider. Determine the number of ways in which Elisa can order these k pints. The order in which Elisa orders matters. Nick is not only your friendly TA, he also has a part-time job in a grocery store. This store sells n different types of India Pale Ale (IPA) and n different types of wheat beer, where n greaterthanorequalto 2 is an integer. Prove that (2n 2) = 2 (n 2) + n^2 by counting, in two different ways, the number of ways to choose two different types of beer. A string consisting of characters is called cool, if exactly one character in the string is equal to x and each other character is a digit. Let n greaterthanorequalto 1 be an integer. Determine the number of cool strings of length n. Let k be an integer with 1 lessthanorequalto k lessthanorequalto n. Determine the number of cool strings of length n that contain exactly n - k many 0's. Use the above two results to prove that sigma_k=1^n k (n k) g^k-1 = n middot 10^n-1. Determine the number of elements x in the set {1,2,3,...,99999} for which the sum of the digits in the decimal representation of r is equal to 8. An example of such an element x is 3041. You may use any result that was proven in class. Let n greaterthanorequalto 2 be an integer and let G = (V, E) be a graph whose vertex set V has size n and whose edge set E is non-empty. The degree of any vertex u is defined to be the number of edges in E that contain u as a vertex.^1 President of the Carleton Computer Science Society

Explanation / Answer

Question 5:

The Number of ways Elisa can order l pints out of n cider = nCl

Now, the number of pints left = k-l

The Number of ways Elisa can order k-l pints out of m beer = mCk-l

These k pints can be ordered in k! number of ways.

Answer: Hence, total number of ways to order k pints =  nCl * mCk-l * k!

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