1) Give a combinatorial proof of the following identity (0 <k<n): n2 k ---- = n.29-1 ke=0
Q1: Give a bijective proof of the following identity E (%) = 2:1 using the following steps: (a) Use binomial paths represented as sequences (or tuples) of steps to define a set 24 representing the left-hand side. (b) Define a set Nr representing the right-hand side as a certain set of tuples with entries from the set {0,1}. (c) Define a bijective function I : 124 + S2R. (d) Show that your function in (c) is well defined. (There is...
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.
9. Give a proof of the following identity using a double-counting argument: -k) = (m+n) Then using this result, derive the following special case from it. This can be done algebraically in just a few steps (you don't need to give a separate counting argument for this): Ëm*- (1)
Let a and b be positive integers, and k an integer with O k<ab. Provide a bijective proof for the following identity
use a bijective argument 1 k/n) m-1 Prove that n2n-l-Li
(1) Using the identity: n n! (2) want k k!(n - k)! for n > 1, prove the following identity: ()-20) + n2
1. Provide a complete and accurate e-N proof that the following sequences converge. That is, prove these sequences converge n-+2 (b) an= n-cos(n) 4n2-7 Tn (d) { } 2. Prove that the following sequences diverge. (Def 7.10 pg 596) READ Sequences that Diverge to oo or-oo (b) ann infinity. Hint: Provide an M -N proof that an approaches 1. Provide a complete and accurate e-N proof that the following sequences converge. That is, prove these sequences converge n-+2 (b) an=...
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.)
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...