6.1 MTH 288

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider a set A with five elements. Find the number of one-to-one functions from the set A to a set with 2 elements.

0

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. An office building contains 30 floors and has 37 offices on each floor. How many offices are in the building? NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. An office building contains 27 floors and has 37 offices on each floor. How many offices are in the building?

1110 999 Since there are 27 floors and 37 offices on each floor, the product rule is applied to obtain 27 ⋅ 37 offices in total.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the positive integers less than 1000. How many positive integers less than 1000 are divisible by both 7 and 11? (You must provide an answer before moving to the next part.)

12

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. A multiple-choice test contains 13 questions. There are four possible answers for each question. In how many ways can a student answer the questions on the test if the student can leave answers blank? NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. A multiple-choice test contains 11 questions. There are four possible answers for each question. In how many ways can a student answer the questions on the test if the student can leave answers blank?

1220703125 48828125 Since there are 11 tasks and five ways to do each task, the product rule applies and yields that there are 5 · 5 · · ·5 = 511 ways a student can answer the questions.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider people with three-letter initials. How many different three-letter initials are there with no letters repeated?

15600

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider a set A with five elements. How many one-to-one functions are there from the set A to a set with 7 elements? NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider a set A with five elements. How many one-to-one functions are there from the set A to a set with 13 elements?

2520 154440 The first choice can be made in 13 ways, since any element of the codomain can be the image of the first element of the domain. After that choice has been made, there are only 12 elements of the codomain available to be the image of the second element of the domain, since images must be distinct for the function to be one-to-one. Similarly, for the third element of the domain, there are 11 choices for a function value. Continuing in this way and applying the product rule, we see that there are 13 ⋅ 12 ⋅ 11 ⋅ 10 ⋅ 9 one-to-one functions from the set A to a set with 13 elements.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider people with three-letter initials. What are the number of choices for the first-, second-, and third-letter initials, if none of the letters are repeated? (You must provide an answer before moving to the next part.)

26, 25, and 24

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider all strings of three decimal digits. How many strings of three decimal digits have exactly two digits that are 4s?

27

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the positive integers less than 1000. How many positive integers less than 1000 have distinct digits and are even?

373

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider all strings of three decimal digits. How many strings of three decimal digits begin with an odd digit? (You must provide an answer before moving to the next part.)

500

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. A multiple-choice test contains 13 questions. There are four possible answers for each question. In how many ways can a student answer the questions on the test if the student answers every question? NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. A multiple-choice test contains 11 questions. There are four possible answers for each question. In how many ways can a student answer the questions on the test if the student answers every question?

67108864 4194304 Since there are 11 tasks and four ways to do each task, the product rule applies and yields that there are 4 · 4 · · ·4 = 411 ways a student can answer the questions.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider a set A with five elements. Find the number of one-to-one functions from the set A to a set with 8 elements. NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider a set A with five elements. Find the number of one-to-one functions from the set A to a set with 5 elements.

6720 120 The first choice can be made in 5 ways, since any element of the codomain can be the image of the first element of the domain. After that choice has been made, there are only 4 elements of the codomain available to be the image of the second element of the domain, since images must be distinct for the function to be one-to-one. Similarly, for the third element of the domain, there are 3 choices for a function value. Continuing in this way and applying the product rule, we see that there are 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 one-to-one functions from the set A to a set with 5 elements.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider a set A with five elements. How many one-to-one functions are there from the set A to a set with 6 elements?

720

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the positive integers less than 1000. How many positive integers less than 1000 are divisible by neither 7 nor 11? (You must provide an answer before moving to the next part.)

779

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider all strings of three decimal digits. How many strings of three decimal digits do not contain the same digit three times? (You must provide an answer before moving to the next part.)

990

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the positive integers less than 1000. Which rule must be used to find the number of positive integers less than 1000 that are divisible by 7? (You must provide an answer before moving to the next part.)

the division rule of counting

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the positive integers less than 1000. Identify the rules used to find the number of positive integers less than 1000 that are divisible by exactly one of 7 and 11. (Check all that apply.) (You must provide an answer before moving to the next part.)

the principle of inclusion-exclusion for sets the division rule

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the positive integers less than 1000. Identify the rules used to find the number of positive integers less than 1000 that are divisible by either 7 or 11. (Check all that apply.) (You must provide an answer before moving to the next part.)

the principle of inclusion-exclusion for sets the division rule for counting

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. An office building contains 30 floors and has 37 offices on each floor. Which rule must be used to find the total number of offices in the building? (You must provide an answer before moving to the next part.)

the product rule

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the positive integers less than 1000. Identify the rules used to find the number of positive integers less than 1000 that have distinct digits. (Check all that apply.) (You must provide an answer before moving to the next part.)

the product rule the sum rule

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the positive integers less than 1000. Which of the following rules is used to find the number of positive integers less than 1000 that are divisible by 7 but not by 11? (Check all that apply.) (You must provide an answer before moving to the next part.)

the subtraction rule the division rule