countinggggg

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

C(10,1) + C(10,3) + C(10, 5) + C(10, 7) + C(10, 9)

How many different permutations are there of the set {a, b, c, d, e, f, g}

P(7,7)

How many ways are there to select a first-prizewinner, a second prize winner, and a third-prize winner from 100 different people who have entered a contest?

100! / (100-3)!

A student can choose a computer project from one of three lists. The three lists contain 23, 15, and 19 possible projects, respectively. No project is on more than one list. How many possible projects are there to choose from?

23 + 15 +19

How many different three-letter initials can people have

26^3

How many permutations of the letters ABCDEFGH contain the string ABC ?

ABC is one object D E F G H 6!

Permutation

Permutation is ORDERD arangement of objects

How many ways are there to select five players from a 10-member tennis team

10! / 5! 5!

A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many different types of this shirt are made?

12x2x3

There are 18 mathematics majors and 325 computer science majors at a college. In how many ways can one representative be picked who is either a mathematics major or a computer science major?

18 + 35

There are 18 mathematics majors and 325 computer science majors at a college. In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer science major?

18x325

There is one position available for a faculty job at Irvine. The applicant must come from either Berkeley which has 20 candidates or UCLA which has 50 candidates. What is the total number of possible candidates for the position

20 + 50

Suppose statement labels in a programming language can be either a single letter or a letter followed by a digit. Find the number of possible labels

26 + 26 . 10

How many ways can we label a chair if each label consists of both a letter AND a number between 1 and 50, inclusive? There are 26 possibilities for the letter AND 50 for the number.

26 x 50

How many different license plates can be made if each plate contains a sequence of three uppercase English letters followed by three digits?

26 ∙ 26 ∙ 26 ∙ 10 ∙ 10 ∙ 10 = 17,576,000

How many different three-letter initials are there that begin with an A?

26^2 (since one letter is fixed)

How many license plates can be made using either two uppercase English letters followed by four digits or two digits followed by four uppercase English letters?

26^2 x 10^2 + 26^2 x 10^4

How many bit strings of length ten both begin and end with

2^8

With 31 flavors of ice cream, 4 sizes of serving, and a choice of "cone" or "dish," how many different orders of ice cream are there?

31 x 4 x 2

The mathematics department must choose either a student or a faculty member as a representative for a university committee. How many choices are there for this representative if there are 37 members of the mathematics faculty and 83 mathematics majors and no one is both a faculty member and a student

37 + 83

A committee is formed consisting of one representative from each of the 50 states in the United States, where the representative from a state is either the governor or one of the two senators from that state. How many ways are there to form this committee?

3^50

A multiple-choice test contains 10 questions. There are four possible answers for each question. a) In how many ways can a student answer the questions on the test if the student answers every question? b) In how many ways can a student answer the questions on the test if the student can leave answers blank?

4^10 Here, blank is an option in addition to the original four possible answers. Hence, the solution is 5^10

How many hands of five cards can be dealt from a standard deck of 52 cards?

52! / 5! (52-5)!

How many ways are there to select 47cards from a deck of 52cards

52! / 5! (52-5)!

How many permutations of {a, b, c, d, e, f, g} end with a?

6! @@@?

Suppose that a saleswoman has to visit eight different cities. She must begin her trip in a specified city, but she can visit the other seven cities in any order she wishes. How many possible orders can the saleswoman use when visiting these cities?

7!

Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}. What is |A ∪ B|?

A ∪ B = {1, 2, 3, 4, 5, 6, 7}. By inspection, |A ∪ B| = 7, but you can also verify that |A ∪ B| = |A| + |B| − |A ∩ B| = 5 + 5 − 3 = 7

Suppose that a department contains 10 men and 15 women. How many ways are there to form a committee with six members if it must have the same number of men and women?

C(15,3) X C(10,3)

In how many ways can we select 5 letters from the english alphabet?

C(26,5)

Product rule

If there are n1 ways to do the first task, and n2 ways to do the second, there are n1⋅n2 ways to do the procedure

How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with 12 horses if all orders of finish are possible

P(12,3)

How many bit strings of length seven are there?

Since each of the seven bits is either a 0 or a 1, the answer is 27 = 128.

How many bit strings of length eight either start with a 1 bit or end with the two bits 00?

Start with 1: 2^ 7 End with 00: 2^ 6 Start with 1 AND end with 0: 2^ 5 2^7 + 2^6 - 2^5

Sum rule

Task 1 can be done n1 ways OR t2 can be done in n2 ways n1+n2 ways to do the task

Combination

UNIRDERED selection

How many functions are there from a set with m elements to a set with n elements?

m ^ n from power to

How many one-to-one functions are there from a set with m elements to one with n elements?

n(n-1)(n-2)...(n-m+1)

How many bitstrings of length eight either begin with 00 or end with 10

There are 26 that begin with 00, 25 that end with 101, 23 that start with 00 and end with 101. So the number of bitstrings with at least one of the two properties is 26 + 25 − 23 = 88

How many positive integers not bigger than 20 are divisible by either 2 or 3

There are ⌊20/2⌋ = 10 that are divisible by 2, and ⌊20/3⌋ = 6 that are divisible by 3. But there are also ⌊20/6⌋ = 3 that are divisible by both 2 and 3, so the total is 10 + 6 − 3 = 13.

List all the permutations of {a, b, c}.

abc, acb, bac, bca, cab, cba

Each user on a computer system has a password, which is six to eight characters long, where each character is an uppercase letter or a digit. Each password must contain at least one digit. How many possible passwords are there

ans

Subtraction Rule

one of n1 ways or in one of n2 ways, then the total number of ways to do the task is n1 + n2 minus the number of ways to do the task that are common to the two different ways

In how many different orders can five runners finish a race if no ties are allowed?

we will use 5 runners from the 5 runners, So, P(5, 5) = 5! / (5-5)! = 5! / 0! = 120