An identifier consists of 2 letters chosen from {m, n, p, r, s} followed by 3 digits from 0–9.  (1) How many identifiers are possible if letters and digits may repeat? [a1] (2) How many identifiers are possible if letters may repeat but digits may not repeat? [a2] (3) How many identifiers containing at least one repeated letter or repeated digit? [a3] (Please fill each blank with an integer and without comma or period.)

A bit string is a finite sequence of 0’s and 1’s. (1) How many bit strings have length 9? [a1] (2) How many bit strings of length 9 begin with five zeros? [a2] (3) How many bit strings of length 9 begin with two zeros and end with two ones? [a3] (Please fill each blank with an integer, without commas or periods.) 

(3 points for the correct answer. 1 bonus point if you have correctly answered Questions 1 through 5.) A survey was conducted in a class of 50 students regarding the math and computer science courses they had taken prior to the current class. The results are as follows: (1) 30 students took Precalculus, (2) 18 students took Calculus, (3) 26 students took Data Mining, (4) 9 students took both Precalculus and Calculus, (5) 16 students took both Precalculus and Data Mining, (6) 47 students took at least one of the three courses, (7) 6 students took all three courses. How many students took both Calculus and Data mining? Hint: Consider the inclusion/exclusion rule for three sets: If A, B, and C are any finite sets, then    

A basketball player makes each free throw with probability 70%, independently. (1) What is the probability he makes exactly one of his next three free throws? [a1] (2) What is the probability he makes at least two shots of his next three free throws? [a2] (3) What is the probability he makes all three free throws? [a3]  (Please enter your answer as a decimal number in the form 0.xxx, keeping three digits after the decimal point.)

A bit string is a finite sequence of 0’s and 1’s. (1) How many bit strings have length 9? [a1] (2) How many bit strings of length 9 begin with four zeros? [a2] (3) How many bit strings of length 9 begin with 1 and end with two ones? [a3] (Please fill each blank with an integer and without commas or periods.) 

What should Trendy’s optimal order size be to start the second half of the season?

What is the maximum gain that Trendy could achieve by eliminating all over- and understock over the entire season?

What is Trendy’s expected profit for the first half of the season?

What is the expected profit (highest or best-case) of Basic over the entire season?           (1st season + 2nd season)?

Please find below the exam file. You can download this word file, provide your solutions, then upload the word file with solutions. Alternatively, if writing on paper, you can take pictures and combine them into a single word or PDF file for submission. The exam is close-book close-note. Make-up Exam.docx