Exam 6: A: Counting

You have 20 cards and 12 envelopes (labeled 1, 2, . . . , 12). In how many ways can you put the 20 cards into the envelopes if (a) the cards are distinct. (b) the cards are identical.

suppose you have 30 books (15 novels, 10 history books, and 5 math books). Assume that all 30 books are different. -In how many ways can you put the 30 books in a row on a shelf?

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words have exactly one vowel?

Suppose | $A$ | =4 and | $B$ | =10 . Find the number of 1-1 functions $f : A \rightarrow B$

(a) Find the number of solutions to $x + y + z = 32$ , where $x , y$ ,and $Z$ are nonnegative integers. (b) Answer part (a), but assume that $x \geq 7$ and $y \geq 15$

A computer is programmed to print subsets of {1, 2, 3, 4, 5} at random. If the computer prints 40 subsets, prove that some subset must have been printed at least twice.

In a technician's box there are 400 VLSI chips, 12 of which are faulty. How many ways are there to pick two chips, so that one is a working chip and the other is faulty? (Assume that no chips are identical.)

Each user has a password 6 characters long where each character is an uppercase letter, a lowercase letter, or a digit. Each password must contain at least one digit. How long will it take to check every possible character combination, if each check takes one unit of time.

assume that you have a bowl containing hard candies: 50 cherry, 50 strawberry, 40 orange, 70 lemon, and 40 pineapple. Assuming that the pieces of each flavor are identical, -How many handfuls of 15 are possible with at least two pieces of each flavor?

How many permutations of the seven letters A, B, C, D, E, F, G are there?

suppose you have a class with 30 students-10 freshmen, 12 sophomores, and 8 juniors. -In how many ways can you put all 30 in a line?

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8.

Find the number of permutations of the letters in the word CORRECT.

suppose you have 30 books (15 novels, 10 history books, and 5 math books). Assume that all 30 books are different. -In how many ways can you put the 30 books in a row on a shelf if the five math books are to be grouped together, but there are no restrictions on the placement of the other books?

nine people (Ann, Ben, Cal, Dot, Ed, Fran, Gail, Hal, and Ida) are in a room. Five of them stand in a row for a picture. -In how many ways can this be done if neither Ed nor Fran are in the picture?

A professor teaching a Discrete Math course gives a multiple choice quiz that has ten questions, each with four possible responses: a, b, c, d. What is the minimum number of students that must be in the professor's class in order to guarantee that at least three answer sheets must be identical? (Assume that no answers are left blank.)

suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words begin with a vowel or end with a vowel?

How many permutations of the seven letters A, B, C, D, E, F, G have the two vowels before the five conso-

A computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of computers.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly three elements, all of them even.