[Q1] [20 points] You can leave answers unsimplified. (A) How many lists of length 5 can be made from the letters A, B, C, D, E, F, G. H if repetition is allowed? (B) How many lists of length 5 can be made from the letters A, B, C, D, E, F, G, H if repetition is not allowed? (C) How many lists of length 5 can be made from the letters A, B, C, D, E, F, G, H if repetition is not allowed and the list must contain the letter BY (D) Suppose you need to choose 3 people from a group of 7 people to go on a trip to Hawaii. How many possibilities of different groups are there? (E) Suppose you are forming a 6-person committee from a group of 10 biology student, 10 math students, and 10 physics students. How many possibilities are there if your committee needs to have exactly 3 biology students, 2 math students, and 1 physics student?[Q2] [20 points] CHOOSE ONE of the following two statements, and prove it using a DIRECT PROOF. You need to clearly mark your choice. If you prove both statements, you will receive 3 extra credit points. Write every step of your proof clearly. (A) If n is an odd integer, then n' + n is even. (B) If cla and cub, then c (a -63).[Q3] [20 points] CHOOSE ONE of the following two statements, and prove it using a CONTRA- POSITIVE PROOF. You need to clearly mark your choice. If you prove both statements, you will receive 3 extra credit points. Write every step of your proof clearly. (A) Suppose n E Z, if 5n + 4 is odd, then n is odd. (B) Suppose r, y ( R, ifr ty > 2, then r 2 1 or y > 1.[Q4] [20 points] CHOOSE ONE of the following two statements, and prove it hy CONTRADIC- TION. You need to clearly mark your choice. If yvoun prove both statements, yon will receive 3 extra credit points. Write overy step of your proof clearly. (A) Suppose n Z, if n* + 5 is odd, then n is even. (B) Suppose a,b,c Z if a|band a{ e, then af(b+c) [Q5] [10 points] CHOOSE ONE of the following two statements, and prove it by INDUCTION. You need to clearly mark your choice. If you prove both statements, you will receive 3 extra credit points. Write every step of your proof clearly. (A) 1+3+3'+3 +...+37-1- 30 - 1 2 (B) n! > 2" for n 2 4.[Q6] [10 points] Consider the following true statement: "If f(x) is differentiable at r = a then f(x) is continuous at s = q." Is the reasoning below, correct or incorrect? Why or why not? (A) The function f(r) = is not continuous at a = 1, therefore it cannot be differentiable at s = 1. T-1 (B) The function /(r) = r-2 is not differentiable at r = 2, therefore it cannot be continuous at r = 2