Exam 6: Counting

In how many ways can you put the 30 books in a row on a shelf if the novels are on the left, the math books are in the middle, and the history books are on the right?

assume that you have 50 pennies and three jars, labeled A, B, and -In how many ways can you put the pennies in the jars, assuming that the pennies are identical and each jar must have at least two pennies put into it?

Using the ordinary alphabet and allowing repeated letters, find the number of words of length 8 that begin and end with the same letter.

In questions suppose you have a class with 30 students - 10 freshmen, 12 sophomores, and 8 juniors. -In how many ways can you get a committee of 4 freshmen and 3 sophomores?

Let $s _ { 1 } , s _ { 2 } \ldots , s _ { 101 }$ be 101 bit strings of length at most 9. Prove that there exist two strings, $s _ { i }$ and $s _ { j }$ where $i \neq j$ that contain the same number of $0 \text { 's }$ and the same number of $\text { 1's. }$ (For example, strings 001001 and 101000 contain the same number of $\text { 0's }$ and the same number of $\text { 1's. }$ )

In how many ways can you put the 30 books in a row on a shelf?

Use the binomial theorem to prove the following: $\left( \begin{array} { l } 6 \\0\end{array} \right) + \left( \begin{array} { l } 6 \\1\end{array} \right) + \cdots + \left( \begin{array} { l } 6 \\6\end{array} \right) = 2 ^ { 6 }$

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?

In questions 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 both Ed and Gail are in the picture?

A club with 20 women and 17 men needs to choose three different members to be president, vice president, and treasurer. (a) In how many ways is this possible? (b) In how many ways is this possible if women will be chosen as president and vice president and a man as treasurer?

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

Find the number of subsets of S = {1, 2, 3, . . . , 10} with exactly five elements, two of which are 3 and 4.

Suppose a restaurant serves a "special dinner" consisting of soup, salad, entree, dessert, and beverage. The restaurant has five kinds of soup, three kinds of salad, ten entrees, five desserts, and four beverages. How many different special dinners are possible? (Two special dinners are different if they differ in at least one selection.)

assume that you have 50 pennies and three jars, labeled A, B, and -In how many ways can you put the pennies in the jars, assuming that the pennies are identical?

How many ways are there to select 6 students from a class of 25 to hold six different executive positions on a committee?

A doughnut shop sells 30 kinds of doughnuts. In how many ways can you (a) get a bag of 12 doughnuts? (b) get a bag of 12 doughnuts if you want at least 3 glazed doughnuts and at least 4 raspberry doughnuts? (c) get a bag of 12 doughnuts if you want exactly 3 glazed doughnuts and exactly 4 raspberry doughnuts?