can you help with these. For question 8 fill in the proof in the right order using the letters below (use all letters in proper order).

Assignment 7 W20: Problem 7 Previous Problem Problem List Next Problem (1 point) Use the theorem that (T) = ("TZ) + ("!) for all m > 1 > 0, and the principle of mathematical induction to prove the formula (6 + (") + (2) +...+ (") = 2" for all non-negative integers n. NOTE: Input your answers, when appropriate, in the following manner: i.e. (*) is equivalent to entering C(n, k) into the answer box. Proof: Base Case: (n = 0) Thus the base case holds for n = 0. Inductive Hypothesis: Suppose the formula is true for n = k 2 ; that is (") + (1) + (2) +...+ ( # )- Inductive Step: We shall verify the formula for n = ; that is (*$1 ) + (*+1) + (121 ) + ... + Using the theorem (7) = ("T_) + (", ]) for all m > 1 > 0, we obtain that ( 4 $1 ) + (4 + 1 ) + ( 1: 21 ) + ... + (1 # 1 ) + (2+1 )= 1+1 + +...+[ ++1. Rearranging the terms to group odd-numbered and even-numbered terms together separately, we have: ( 4: $ 1 ) + ( 1 71 ) + ( 1521) + ... + (1: #1 ) + (*+1)= [1+ +... + + + + ... + +1 = + - 2*+1 ( by the inductive hypothesis ). Q. E. D.Assignment 7 W20: Problem 8 Previous Problem Problem List Next Problem (1 point) Prove the following using the definition of (") Proof: RS = (2+7) + ("#2 ) = V V V = LS Q. E. D. A. (n + 2)!(n + 3) (n - k + 3)!k! B (n + 3)! (n + 3 - k)!k! C. (n + 2)! n + 2)! (n - k + 3)!(k - 1)! + (n + 2 - k)!k! D. (n+3) E . . k(n + 2)! + (n - k+ 3)(n+ 2)! (n - k + 3)!k! F. (n + 2) !(k + n - k+ 3) (n - k + 3)!k