(3) Using the identity: (*) – 16–191 n! k!(n-k! k for n > 2, prove the...
(2) Using the identity: n! k!(n - k)! for n > 2, prove that the following identity is even: 1 n
(1) Using the identity: n n! (2) want k k!(n - k)! for n > 1, prove the following identity: ()-20) + n2
Use induction and Pascal's identity to prove that (7) = 2" where n > 0.
1) Give a combinatorial proof of the following identity (0 <k<n): n2 k ---- = n.29-1 ke=0
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.
2. Use induction to prove that the following identity holds for al k 2 (n 1)2"+12 Be sure to clearly state your induction hypothesis, and state whether you're using weak induction or strong induction
(c) contrapositive positiv 2. (a) Prove that for all integers n and k where n >k>0, (+1) = 0)+2). (b) Let k be a positive integer. Prove by induction on n that ¿ () = 1) for all integers n > k. 3. An urn contains five white balls numbered from 1 to 5. five red balls numbered from 1 to 5 and fiv
2. Use Method of mathematical induction to prove identity : for all natural n > 2 1.1+(1.1)? + ... + (1.1)n-1 = - 11n-1 1.1 - (1.1)" - 0.1 inf of the set below
.n= n(n-1)(n+1) for all n > 2. 12. Use induction to prove (1 : 2) +(2-3)+(3-4) +...+(n-1).n [9 points) 3
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.