离散结构小考助攻 CS 2100代写

CS 2100: Discrete Structures

Quiz 5

1.(10 points)

Consider the set A = {1, 2, 3, 4, 5, 6, 7}.

(a) How many 4 element subsets are there of A?

(b) How many 4 element subsets of A are there that contain 4 or 6?

Note: calculate the final integer results! To receive partial credit, write down all necessary steps of your reasoning.

2.(20 points) 离散结构小考助攻

(a) How many different ways are there to form 7 digit strings having no 0’s where we only distinguish between the number of appearances of each digit. For example, 7777555 and 5557777 count the same because both contain four 7’s and three 5’s; 5557777 and 5547777 are distinct because the first number has three 5’s and no 4’s, whereas the second number has two 5’s and one 4.

(b) Answer the problem above but with the constraint that each instance can have no more than two 7’s. [Hint: rule of complements.]

Note: express your answer in terms of formulas such as P(n, r), C(n, r), n^r or n!. To receive partial credit, write down all necessary steps of your reasoning.

3.(20 points) 离散结构小考助攻

30 children from the local elementary school, fifteen 4th-graders and fifteen 5th-graders, are standing in line. Assume that all the possible ways the children might line up are equally likely.

(a) Assuming no two of the children have the same name, what is the probability the children appear in line in alphabetical order by name?

(b) What is the probability all the 5th-graders precede the 4th-graders?

Note: express your answers as fractions of formulas such as P(n, r), C(n, r), n^r, n! or as small numbers. To receive partial credit, write down all necessary steps of your reasoning.

4.(20 points) 离散结构小考助攻

A newbie hacker wants to break a security system of an internet company. She faces a password consisting of 6 characters drawn from the letters A–Z (26 letters) and 0–9 digits (10 digits). Characters can be repeated in a password. Note that both letters and digits are characters, and no lower-case letters are present.

(a) How many passwords are possible?

(b) According to intelligence data, the password contains at least one digit (0-9). How many passwords are possible in this case?

(c) It is well known that ‘1234’ is a common sequence used in passwords. How many passwords are possible if they have to contain the sequence ‘1234’? Note that these characters must occur in exactly this sequence; for example ‘1A234’ is not an example of this sequence.

(d) How many possible passwords contain at least one duplicate character?

Note: express your answer in terms of formulas such as P(n, r), C(n, r), n^r or n!. To receive partial credit, write down all necessary steps of your reasoning.

5.(10 points) 离散结构小考助攻

Consider a bag of 10 marbles, with 6 red marbles and 4 blue marbles. Suppose I take two marbles from the bag at once (without looking), and I let the first one be called M1 and the second one M2.

(a) Suppose I show you M1 and it is red. What is the probability that M2 is red?

(b) Suppose I show you M1 and it is blue. What is the probability that M2 is red?

Note: calculate the final results as fractions of integers. To receive partial credit, write down all necessary steps of your reasoning.

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