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PLEASE DO BOTH
(5) (5 pts) Prove the identity (1) () = (x) (---), whenever n,...
PLEASE DO BOTH (9) (5 pts) How many solutions are there to the inequality 11 +*2+I3...
PLEASE DO BOTH
(9) (5 pts) How many solutions are there to the inequality 11 +*2+I3 <11, where 21, 22, and 13 are non-negative integers? Hint: Introduce an auxiliary variable In such that 21+12+13+14 = 11. (10) (5 pts) How many terms are there in the expansion of (2x + 3y + 5z)50?
please do both 4 and 5 and show work! will upvote fast Problem VII.4. How many...
please do both 4 and 5 and show work! will upvote fast
Problem VII.4. How many non-negative integer vectors are there such that where exactly k of the entries zero? N Problem VII.5. Prove the following: () ik
5. Prove each of the following set equalities both by Venn Diagram and by algebraic method....
5. Prove each of the following set equalities both by Venn
Diagram and by algebraic method.
(a) A - (B C) = (A - B)
(A - C)
(b) A - (B C) = (A - B)
(A - C)
(c) A (B - C) = (A
B) - C = (A
B) - (A C)
Hint: To prove the last form, use the equality
A C' =
A (A'
C').
(d) A (B - C) = (A
B) (A...
Problem 5 5.a Consider the following identity. For all positive integers n and k with n...
Problem 5 5.a Consider the following identity. For all positive integers n and k with n 2k, (n choose k) + (n choose k-1) = (n+1 choose k). This can be demonstrated either algebraically or via a story proof. To prove the identity algebraically, we can write (n choose k) + (n choose k-1) = n!/[k!(n-k)!] + n!/[(k-1)!(n-k+1)!] = [(n-k+1)n! + (k)n!]/[k!(n-k+1)!] [n!(n+1)/k!(n-k+1)!] = (n+1 choose k). Which of the following is a story proof of the identity? Consider a...
Please send me solutions for the above five questions. The questions are based on Pigeonhole Principle. 3. A shop co...
Please send me solutions for the above five
questions.
The questions are based on Pigeonhole Principle.
3. A shop contains twelve samples of read shirts, seven samples of white shirts, and N samples of blue shirts. Suppose that the smallest K such that choosing K samples from the collection guarantees that you have six samples of the same color of shirt is K-15. What is N? 4. Show that among any n1 positive integers not exceeding 2nthere must be integer...
Optional Extra Points (20 points) (a) [5 points Suppose that k and n are integers with 1 Sk
Optional Extra Points (20 points) (a) [5 points Suppose that k and n are integers with 1 Sk<n. Prove the hexagon identity which relates terms n Pascal's triangle that form a hexagon A circular r-permutation of n people is a seating of r of thesen people around a circular table, where seatings are considered to be the same if they can be obtained from each other by rotating the table. (b) [3 points Find the number of circular 3-permutations of...
PLEASE DO BOTH (7) (5 pts) A croissant shop has plain croissants, cherry croissants, chocolate croissants,...
PLEASE DO BOTH
(7) (5 pts) A croissant shop has plain croissants, cherry croissants, chocolate croissants, almond croissants, apple croissants, and broccoli croissants. How many ways are there to choose (a) a dozen croissants? (b) two dozen croissants with at least two of each kind? (8) (5 pts) How many solutions are there to the equation 11 +12+13 +In+ 15 + 16 = 29, where I, for i = 1,2,3,4,5,6, is a non-negative integer such that (a) Ii > 1...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
PLEASE DO BOTH (1) (5 pts) How many ways can you permute the letters in BANANA?...
PLEASE DO BOTH
(1) (5 pts) How many ways can you permute the letters in BANANA? (2) (5 pts) At a Chinese restaurant, dinner for 8 people consists of 3 items from column A, 4 items from column B and 3 items from column C. If columns A, B and C have 5, 7 and 6 items respectively how many different dinner combinations are possible?
prove that A is non singular 5.(25 pts) For each positive integer n, let f()(+2)(1)(0,1. Let...
prove that A is non singular
5.(25 pts) For each positive integer n, let f()(+2)(1)(0,1. Let f()-0, (1) Prove that (fn) converges to fpointwisely on (0, 1) (2) Does (n) converges to f uniformly on (0, 1]?
PLEASE DO BOTH
(9) (5 pts) How many solutions are there to the inequality 11 +*2+I3 <11, where 21, 22, and 13 are non-negative integers? Hint: Introduce an auxiliary variable In such that 21+12+13+14 = 11. (10) (5 pts) How many terms are there in the expansion of (2x + 3y + 5z)50?
please do both 4 and 5 and show work! will upvote fast
Problem VII.4. How many non-negative integer vectors are there such that where exactly k of the entries zero? N Problem VII.5. Prove the following: () ik
5. Prove each of the following set equalities both by Venn
Diagram and by algebraic method.
(a) A - (B C) = (A - B)
(A - C)
(b) A - (B C) = (A - B)
(A - C)
(c) A (B - C) = (A
B) - C = (A
B) - (A C)
Hint: To prove the last form, use the equality
A C' =
A (A'
C').
(d) A (B - C) = (A
B) (A...
Problem 5 5.a Consider the following identity. For all positive integers n and k with n 2k, (n choose k) + (n choose k-1) = (n+1 choose k). This can be demonstrated either algebraically or via a story proof. To prove the identity algebraically, we can write (n choose k) + (n choose k-1) = n!/[k!(n-k)!] + n!/[(k-1)!(n-k+1)!] = [(n-k+1)n! + (k)n!]/[k!(n-k+1)!] [n!(n+1)/k!(n-k+1)!] = (n+1 choose k). Which of the following is a story proof of the identity? Consider a...
Please send me solutions for the above five
questions.
The questions are based on Pigeonhole Principle.
3. A shop contains twelve samples of read shirts, seven samples of white shirts, and N samples of blue shirts. Suppose that the smallest K such that choosing K samples from the collection guarantees that you have six samples of the same color of shirt is K-15. What is N? 4. Show that among any n1 positive integers not exceeding 2nthere must be integer...
Optional Extra Points (20 points) (a) [5 points Suppose that k and n are integers with 1 Sk<n. Prove the hexagon identity which relates terms n Pascal's triangle that form a hexagon A circular r-permutation of n people is a seating of r of thesen people around a circular table, where seatings are considered to be the same if they can be obtained from each other by rotating the table. (b) [3 points Find the number of circular 3-permutations of...
PLEASE DO BOTH
(7) (5 pts) A croissant shop has plain croissants, cherry croissants, chocolate croissants, almond croissants, apple croissants, and broccoli croissants. How many ways are there to choose (a) a dozen croissants? (b) two dozen croissants with at least two of each kind? (8) (5 pts) How many solutions are there to the equation 11 +12+13 +In+ 15 + 16 = 29, where I, for i = 1,2,3,4,5,6, is a non-negative integer such that (a) Ii > 1...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
PLEASE DO BOTH
(1) (5 pts) How many ways can you permute the letters in BANANA? (2) (5 pts) At a Chinese restaurant, dinner for 8 people consists of 3 items from column A, 4 items from column B and 3 items from column C. If columns A, B and C have 5, 7 and 6 items respectively how many different dinner combinations are possible?
prove that A is non singular
5.(25 pts) For each positive integer n, let f()(+2)(1)(0,1. Let f()-0, (1) Prove that (fn) converges to fpointwisely on (0, 1) (2) Does (n) converges to f uniformly on (0, 1]?
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