Exam 6: A: Counting

Explain how the Pigeonhole Principle can be used to show that among any 11 integers, at least two must have the same last digit.

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?

A class consists of 20 sophomores and 15 freshmen. The class needs to form a committee of size five. (a) How many committees are possible? (b) How many committees are possible if the committee must have three sophomores and two freshmen?

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.)

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

Use the binomial theorem to expand $( 2 a + b ) ^ { 4 }$

let A be the set of all bit strings of length 10. -How many bit strings of length 10 have more 0's than 1's?

Find the next four largest 4-combinations of the set {1, 2, 3, 4, 5, 6, 7, 8} after {1, 2, 3, 5}.

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

You have 20 pennies, 30 nickels, and 40 dimes. Assume that the pennies are identical, the nickels are identical, and the dimes are identical. In how many ways can you put all the coins in a row?

If the permutations of 1, 2, 3, 4, 5, 6 are written in lexicographic order, with 123456 in position #1, 123465 in position #2, etc., find the permutation immediately before 534126.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly five elements, but neither 3 nor 4.

Find the number of words of length eight of distinct letters of the alphabet so that the words do not have both A and B in them.

Find the number of subsets of S = {1, 2, 3, . . . , 10} that contain exactly five elements, the sum of which is even.

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

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

A factory makes automobile parts. Each part has a code consisting of a letter and three digits, such as C117, O076, or Z920. Last week the factory made 60,000 parts. Prove that there are at least three parts that have the same serial number.

let A be the set of all bit strings of length 10. -How many bit strings of length 10 begin with 1101?

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?

suppose you have a class with 30 students-10 freshmen, 12 sophomores, and 8 juniors. -In how many ways can you put all students in a line so that the freshmen are first, the sophomores are in the middle, and the juniors are at the end?