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Problem 3. (20 pts) (a) (10 pts) Show that the following identity in Pascal's Triangle holds:...
use scheme program The following pattern of numbers is called Pascal's triangle. The numbers at the...
use scheme program
The following pattern of numbers is called Pascal's
triangle.
The numbers at the edge of the triangle are all 1, and each number
inside the triangle is the sum of the two numbers above it.
(pascals 0 0) → 1
(pascals 2 0) → 1
(pascals 2 1) → 2
(pascals 4 2) → 6
(printTriangle 5)
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
[5 marks] Write a procedure...
The Arithmetical Triangle sparalleles arta Blaise Pascal's 1955 work Treatise on the Arithmetical...
The Arithmetical Triangle sparalleles arta Blaise Pascal's 1955 work Treatise on the Arithmetical Triangle contains a collection several results already known about the "Pascal's Triangle" for more than 500 years, as well as applications of the triangle to probability. Among these results are the Hockey-Stick Theorem the fact that the sum of a row is a power of 2, as well as the following fact about ratios: の(k+1) = (k + 1):(n-k) We have actually already seen this fact somewhere...
3. Give a combinatorial proof of the following identity. ("t?) = () + (-1) where n...
3. Give a combinatorial proof of the following identity. ("t?) = () + (-1) where n and k are positive integers with n > k. 5. Give a combinatorial proof of the following identity (known as the Hockey Stick Identity). (%) + (**") + (**?) + ... + ( )= (#1) where n and k are positive integers with n > k.
(iv) Check that the formula () = 21" holds for rows n = 0 to 5...
(iv) Check that the formula () = 21" holds for rows n = 0 to 5 in Pascal's triangle. (If it doesn't work for n = 4 or 5, go back and redo (iii)!) (v) Prove the formula of (iv) using (ii). (a) using (ii); (b) by proving that both sides of the formula represent the number of subsets of a set of n elements. For the left side use the addition rule for counting after partitioning the collection of...
8.14. Give a combinatorial proof of the following binomial identity: 3 0) (+) = ---() =...
8.14. Give a combinatorial proof of the following binomial identity: 3 0) (+) = ---() = 2n-m k= m (Hint: Think of the number of ways of picking two disjoint subsets from a set of cardinality n so that one subset is of cardinality m and the other subset is arbitrary.)
Question 3* For any n,T EN the biomial coefficient ( is the coefficient of in the expansion of (1...
Question 3* For any n,T EN the biomial coefficient ( is the coefficient of in the expansion of (1 + z)". (E.g. (4) 6 because (1 + z)4-1 + 4x + 612 + 4r' + re) In particular, 0 whenever r >n and ()) for all nEN*. These facts, together with Pascal's identity (")+ )(), facilitate the calculation of the value of () for any particular values of n and r via the well-know 'Pascal's triangle'. a) Use Pascal's identity...
Main topic and problems for the final project The main purpose of the project is to introduce you how to use a in 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...
Therom 1.8.2 n choose k = (n choose n-k) n choose k = (n-1 choose K)...
Therom 1.8.2
n choose k = (n choose n-k)
n choose k = (n-1 choose K) + (n-1 choose K-1)
2n = summation of (n choose i )
please use the induction method
(a) (10 pts) Show that the following equality holds: n +1 + 2 Hint: If you proceed by induction, you might want to use Theorem 1.8.2. If you search for a combinatorial proof, consider the set X - (i,j, k): 0 S i,j< k< n) (b) (10...
Solve and show work for problem 8 Problem 8. Consider the sequence defined by ao =...
Solve and show work for problem 8
Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
Prove by induction that for every positive integer n, the following identity holds: 1+3+5+...+(2n – 1)...
Prove by induction that for every positive integer n, the following identity holds: 1+3+5+...+(2n – 1) = np. Stated in words, this identity shows that the sum of the first n odd numbers is n’.
use scheme program
The following pattern of numbers is called Pascal's
triangle.
The numbers at the edge of the triangle are all 1, and each number
inside the triangle is the sum of the two numbers above it.
(pascals 0 0) → 1
(pascals 2 0) → 1
(pascals 2 1) → 2
(pascals 4 2) → 6
(printTriangle 5)
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
[5 marks] Write a procedure...
The Arithmetical Triangle sparalleles arta Blaise Pascal's 1955 work Treatise on the Arithmetical Triangle contains a collection several results already known about the "Pascal's Triangle" for more than 500 years, as well as applications of the triangle to probability. Among these results are the Hockey-Stick Theorem the fact that the sum of a row is a power of 2, as well as the following fact about ratios: の(k+1) = (k + 1):(n-k) We have actually already seen this fact somewhere...
3. Give a combinatorial proof of the following identity. ("t?) = () + (-1) where n and k are positive integers with n > k. 5. Give a combinatorial proof of the following identity (known as the Hockey Stick Identity). (%) + (**") + (**?) + ... + ( )= (#1) where n and k are positive integers with n > k.
(iv) Check that the formula () = 21" holds for rows n = 0 to 5 in Pascal's triangle. (If it doesn't work for n = 4 or 5, go back and redo (iii)!) (v) Prove the formula of (iv) using (ii). (a) using (ii); (b) by proving that both sides of the formula represent the number of subsets of a set of n elements. For the left side use the addition rule for counting after partitioning the collection of...
8.14. Give a combinatorial proof of the following binomial identity: 3 0) (+) = ---() = 2n-m k= m (Hint: Think of the number of ways of picking two disjoint subsets from a set of cardinality n so that one subset is of cardinality m and the other subset is arbitrary.)
Question 3* For any n,T EN the biomial coefficient ( is the coefficient of in the expansion of (1 + z)". (E.g. (4) 6 because (1 + z)4-1 + 4x + 612 + 4r' + re) In particular, 0 whenever r >n and ()) for all nEN*. These facts, together with Pascal's identity (")+ )(), facilitate the calculation of the value of () for any particular values of n and r via the well-know 'Pascal's triangle'. a) Use Pascal's identity...
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...
Therom 1.8.2
n choose k = (n choose n-k)
n choose k = (n-1 choose K) + (n-1 choose K-1)
2n = summation of (n choose i )
please use the induction method
(a) (10 pts) Show that the following equality holds: n +1 + 2 Hint: If you proceed by induction, you might want to use Theorem 1.8.2. If you search for a combinatorial proof, consider the set X - (i,j, k): 0 S i,j< k< n) (b) (10...
Solve and show work for problem 8
Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
Prove by induction that for every positive integer n, the following identity holds: 1+3+5+...+(2n – 1) = np. Stated in words, this identity shows that the sum of the first n odd numbers is n’.
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