**CS 2100: Discrete Structures **

## Practice Quiz 5

## Fall 2020

Discrete Structures代考 For all problems, express your solutions in terms of factorials, exponents, multiplication,division, addition, and/or subtraction.

*P*(*n, r*) and/or*C*(*n, r*), please providethe actual mathematical formulation.

*P*(10*,*8) as a final answer, write*P*(10*,*8) = 10*×*9 = 90(when the math is straightforward); instead of writing*P*(100*,*50), write at least (when the exact answer is hard to get without a calculator).

### 1.**(16 points) **A suitcase has an ordered, 4-digit combination lock.

(a) How many possible combinations are there?

(b) Counting the number of possible combinations is an example of which of the following?

- set

(c) How many possible combinations have *distinct *digits?

(d) Counting the number of possible combinations with *distinct *digits is an example of which of the following? Discrete Structures代考

(e) How many possible combinations have 7 as the leftmost digit?

(f) How many possible combinations do not have 7 as the leftmost digit?

(g) How many possible combinations have an odd number as the rightmost digit?

(h) How many possible combinations have 7 as the leftmost digit *and *an odd number as the rightmost digit?

(i) How many possible combinations have 7 as the leftmost digit *or *an odd number as the rightmost digit?

### 2.**(10 points) **Suppose that we want to fill a bag with 15 pieces of fruit from a stand that sells 5 types of fruit: apples, bananas, oranges, limes, and lemons.

(a) What is the binary sequence of length 19 that represents a bag of eight apples, zero bananas, two oranges, one lime, and four lemons?

(b) What is the specific sum equation that represents the same bag as in part (a)?

(c) How many ways are there to fill our bag?

(d) How many ways are there to fill our bag such that we get **at least one of each type **of fruit?

### 3.**(5 points) **How many distinguishable arrangements are there of the letters in “INTEGER”? Discrete Structures代考

4.**(10 points) **Consider a club that has eight sophomores, five juniors, and ten seniors as members. Solve the following problems about forming a committee of six students.

(a) How many ways are there to form the committee?

(b) How many ways are there to form the committee if it must have the same number of students from each classification (e.g., sophomores, juniors, seniors)?

(c) How many ways are there to form the committee if it must not have the same number of students from each classification?

(d) What is the probability that a selected committee will have the same number of students from each classification?

### 5.**(6 points) **Suppose that an unfair coin gives heads with probability What is the prob-ability that exactly five heads come up when this coin is flipped eight times? Assume that each of the coins flips is independent. Discrete Structures代考

6.**(8 points) **Suppose that two fair, six-sided dice are rolled, one red and one green.

(a) What is the probability that at least one die shows a 2 or the dice sum to a value no greater than 5?

(b) What is the conditional probability that the value showing for the red die is 2, given the information that the dice sum to a value no greater than 5?

(c) What is the probability that both the value showing for the red die is 2 and the dice sum to a value no greater than 5?

7.**(5 points) **What is the expected amount of money you win in one play of the following game?

Two fair, 10-sided dice are rolled. You win $10 for each 10 that comes up.

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