1) Show all your work in proper mathematical form to receive full marks. An answer with little or no justification may receive no marks. Write 55*54*53*52*51 in the form nPr. 2) Express 100Cso as a sum of two terms from Pascal's triangle. 3) How many arrangements are there of the letters in the word Neutrogena? 4) A newly formed company of 24 partners wants to select a President, Vice-President and Secretary. How many ways can this be done if no one can occupy more than one position? 5) How many five digit numbers are there that start and end with an odd digit and also the middle digit is odd? 6) Give an example of a counting problem where C3 + 4C4 is the solution. 7) Express ssC14 - 54C13 as a single term from Pascal's triangle.8) Determine the number of paths from A to B if you can only move East and South. A B 9) The checker shown below cannot enter the square with an X, but can jump over it. How many paths are there to the top of the board? X X 10) Use Pascal's method to determine the missing terms. 55 230 X Z 380 Y1) Write the expression below in the form (a + b)" (Hint: find a, b and w) m5 + 10man + 40m3n2 + 80 m2n3 + 80mn4 + 32n5 2) Suppose there is a race between 10 friends. Find the probability that Roger is first, George is second and Diep is last? 3) How many arrangements of the letters in the word TORONTO begin with the letter R and end with the letter O? 4) A 12-member jury is to be selected from 22 men and 24 women. Find the probability that this jury has 6 or 7 males.1) Solve for n in the equation. A full algebraic solution is required. E:j: = 110 2} Consider the following set {i}. 1, 2. 3} a} How many four-digit odd numbers can be formed using these digits? Assume repetition is not allowed. b} How many numbers greater than 1.0m can be formed using these digits? Assume repetition is not allowed. 3} How many ways can 9 peple line up if Tony, Billy, and Mary can not be next to each other? 4} Suppose there are it} lawyers. T accountants and 6 computer technicians being considered to form a committee. How many ways can a committee of 6 people be chosen it there must be two lawyers. two accountants and two computer technicians on the committee? 5} Use the Binomial Theorem to expand and simplify the expression (3:? - 5y)'. Show all your work. 6} Suppose you are expanding [3x + Eye)\". a] i) How many terms are there in the expansion? ii] State the last term. in] Without expanding {use the Formulator the General Term) what is the 4th term? Show your work. 5} Suppose a 5 card hand is drawn from a standard deck of cards. Find the probability that this hand contains exactly 2 cards that are divisible by 5 and exactly 2 face cards? 5} An ltirintario license plate has 4 letters followed by 3 numbers. How many license plates a. contain only vowels (A. E, I. {3), U] and numbers greater than 5? Assume repetition is not allowed {example AEIUEBQ}. D. contain 4 different letters but the same 3 digits {example ABCDTEH