Exam 6: A: Counting

How many subsets with an odd number of elements does a set with 10 elements have?

Let $S _ { 1 } , S _ { 2 } \ldots , S _ { 101 }$ be 101 bit strings of length at most 9 . Prove that there exist two strings, $8$ and $S _ { j }$ , where $i \neq j$ , that contain the same number of 0 's and the same number of 1's. (For example, strings 001001 and 101000 contain the same number of 0 's and the same number of 1 's.)

Show that in a group of ten people (where any two people are either friends or enemies) there are either three mutual friends or four mutual enemies.

consider all bit strings of length 12. -How many begin with 110?

How many 8-element DNA sequences start with C and end with C?

a club with 20 women and 17 men needs to form a committee of size six. -How many committees are possible if the committee must have three women and three men?

Find the coefficient of $x ^ { 7 } y ^ { 5 }$ in the expansion of $( 3 x - y ) ^ { 12 }$

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

let A be the set of all strings of decimal digits of length five. For example, 00312 and 19483 are strings in A. -Find |A|.

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?

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

Find the coefficient of $x ^ { 8 } \text { in the expansion of } \left( x ^ { 2 } + 2 \right) ^ { 13 }$

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 or end with A or B?

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?

A computer randomly prints three-digit codes, with no repeated digits in any code (for example, 387, 072, 760). What is the minimum number of codes that must be printed in order to guarantee that at least six of the codes are identical?

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

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 7?

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

How many truth tables are possible for compound propositions with the five variables p, q, r, s, t ?

a club with 20 women and 17 men needs to form a committee of size six. -How many committees are possible if the committee must have at least two men?