Question 1 EREITELIGTY 3 pts Consider the set A of binary strings of length five. Examples are 00000, 01000, 10010, etc. * Let B be a subset of A where all the binary strings in subset B have an even number of 1's. e Let C be a subset of A where all the binary stings in subset C start and end with 1's. e Let D be a subset of A where all the binary strings in subset D have exactly 4 1's. Calculate the size of BU C U D. Hint: Theorem 9.3.3 The Inclusion/Exclusion Rule for Two or Three Sets (Section 9.3, page 596). Question 3 Coursology 3 pts How many distinguishable permutations with repeated elements are there for your 8-digit, numeric student ID?Question 5 UL 3 pts In the game of poker played with an ordinary deck of 52 cards various five-card holdings are given special names. The name "three of a kind" is reserved for a holding that meets the following rule: Three cards of the same denomination and two other cards of different denominations. The number of distinct "three of a kind" that can be drawn from a 52-card deck can be calculated with the following product expression: 13\\ (4 (12 (42 (1) () () Q) Explain what each of the factors in the expression represents in terms of selections from the deck of cards. Follow the examples given in slides 8-10 of the lecture Section 9.5 Combinations.pdf found at Summer Quarter Week 5-1 in Canvas. Question 6 EREJULLIGTY 4 pts Prove that for every integer n where n is greater than or equal to 2, P(n,n) =P(n,n1). For full credit you must, Use the direct proof method. o Explicitly write the universal conditional to expose the hypothesis. Use the factorial definition of a permutation in your proof. Give your proof line-by-line, with each line a statement with its justification. You can use the Canvas math editor or write your math statements in English. For example, the universal statement that is to be proved was written in the Canvas math editor. In English it would be: For all integers n greater than or equal to 2, the permutation of n elements taken n at a time equals the permutation of n elements taken n-1 at a time. Question 7 BEITELTY 2 pts Suppose a group of 24 students consists of fourteen freshmen and ten sophomores. How many six-student teams can be chosen that consist of two freshmen and four sophomores? Give your answer as an integer. 12,600 Question 8 ERELTEILI 3 pts You have a bag of enough ping pong balls such that each one has exactly one of the numbers from 57 to 134 painted on it. What is the probability that if you randomly select five of the balls from the bag all of the numbers on the selected balls will be divisible by either 3 or by 7? Your final answer must be a decimal number rounded to four decimal places. For example, if you calculate the answer 0.166666, your answer would be 0.1667. 0.0046 Question 9 REITELILTW 3 pts For this problem, you need the total, n, of all digits in your student ID. For example, for the student ID 12345678, the total is n = 36. Question: How many solutions, s, are there to the equation 1 + 29 + 3 + x4 + 5 = n ife; 2201