Exam 6: Counting

(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 ≥ 7 and y ≥ 15

Use the binomial theorem to expand $( a + 2 ) ^ { 6 }$

In how many ways can you get a bunch of three history books and seven novels to give to a friend?

In questions let A be the set of all bit strings of length 10. -How many bit strings of length 10 are there?

In questions consider all bit strings of length 12. -How many have exactly four 1's?

List the derangements of 1, 2, 3, 4.

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

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 end with B?

A class consists of 20 sophomores and 15 freshmen. The club needs to choose four different members to be president, vice president, secretary, and treasurer. (a) In how many ways is this possible? (b) In how many ways is this possible if sophomores will be chosen as president and treasurer and freshmen as vice president and secretary?

In questions let A be the set of all bit strings of length 10. -How many bit strings of length 10 have exactly six 0's?

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

In questions consider all bit strings of length 12. -How many begin with 11 and end with 10?

Use the binomial theorem to expand $\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 7 }$

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 Ben is to be in the picture?

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 Dot is on the left end and Ed is on the right end?

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.)

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 one piece of each flavor?

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?

In questions suppose that a "word" is any string of seven letters of the alphabet, with repeated letters allowed. -How many words have no vowels?

In questions 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?