Counting and Probability

The outcome of a random process is sure to occur, but impossible to predict. Examples: fair coin tossing, rolling a pair of dice, card drawing

from a well shuffled card deck, etc. Sample space is the set of all possible outcomes of

a random process. An event is a subset of a sample space. The probability of an event is the ratio between the

number of outcomes that satisfy the event to the total number of possible outcomes.

Possibility Trees

Teams A and B are to play each other repeatedly until one wins two games in a row or a total three games. What is the probability that five games will be

needed to determine the winner? Suppose there are 4 I/O units and 3 CPUs. In

how many ways can I/Os and CPUs be paired with each other when there are no restrictions?

Multiplication Rule

Multiplication rule: if an operation consists of k steps each of which can be performed in ni ways (i = 1, 2, …, k), then the entire operation can be performed in ni ways. Number of PINs with repetitions Number of elements in a Cartesian product Number of PINs without repetition Number of Input/Output tables for a circuit with n input

signals Number of iterations in nested loops

Multiplication Rule

Three officers – a president, a treasurer and a secretary are to be chosen from four people: Alice, Bob, Cindy and Dan. Alice cannot be a president, Either Cindy or Dan must be a secretary. How many ways can the officers be chosen?

Permutations

A permutation of a set of objects is an ordering of these objects.

The number of permutations of a set of n objects is n! .

An r-permutation of a set of n elements is an ordered selection of r elements taken from a set of n elements: P(n, r)

P(n, r) = n! / (n – r)! Show that P(n, 2) + P(n, 1) = n2

Exercises

How many odd integers are there from 10 through 99 that have distinct digits? (40)

How many numbers from 1 through 99999 contain exactly one of each of the digits 2, 3, 4, and 5? (720)

Let n = p1k1p2

k2…pmkm.

How many divisors does n have? What is the smallest integer with exactly 12 divisors? (60)

Addition Rule

If a finite set A is a union of k mutually disjoint sets A1, A2, …, Ak, then N(A) = N(Ai) Number of words of length no more than 3 Number of 3-digit integers divisible by 5

The Difference Rule If A is a finite set and B is its subset, then N(A – B) = N(A) – N(B).

Example 6.3.3: How many 4-symbol PINs contain repeated symbols? (Each symbol is chosen from the 26 letters of the alphabet and the ten digits)

Inclusion/Exclusion Rule

n(AUB) = n(A) + n(B) – n(A∩B). Derive the above rule for 3 sets.

How many integers from 1 through 1000 are multiples of 3 or multiples of 5?

How many integers from 1 through 1000 are neither multiples of 3 nor multiples of 5?

Exercises

Suppose that out of 50 students, 30 know Pascal, 18 know Fortran, 26 know Cobol, 9 know both Pascal and Fortran, 16 know both Pascal and Cobol, 8 know Fortran and Cobol and 47 know at least one programming language. How many students know none of the three languages? (3) How many students know all three languages? (6) How many students know exactly 2 languages? (15)

Exercises

A calculator has an eight-digit display and a decimal point which can be before, after or in between digits. The calculator can also display a minus sign for negative numbers. How many different numbers can the calculator display? (Q. of Hw6)

A combination lock requires three selections of numbers from 1 to 39. How many combinations are possible if the same number cannot be used for adjacent selections? (39*38*38)

Exercises

How many integers from 1 to 100000 contain the digit 6 exactly once / at least once?

What is the probability that a random number from 1 to 100000 will contain two or more occurrences of digit 6?

6 new employees, 2 of whom are married are assigned 6 desks, which are lined up in a row. What is the probability that the married couple will have non-adjacent desks?

Exercises

Consider strings of length n over the set {a, b, c, d}: How many such strings contain at least one pair of

consecutive characters that are the same? If a string of length 10 is chosen at random, what is the

probability that it contains at least on pair of consecutive characters that are the same?

How many permutations of abcde are there in which the first character is a, b, or c and the last character is c, d, or e?

How many integers from 1 through 999999 contain each of the digits 1, 2, and 3 at least once?

Combinations

An r-combination of a set of n elements is a subset of r of the n elements, denoted C(n, r)

or . Permutation is an ordered selection;

combination is an unordered selection. Quantitative relationship between permutations

and combinations: P(n, r) = C(n, r) * r!

See slides later for permutations of a set with repeated elements.

r

n

Team Selection Problems

12 people, 5 men and 7 women, work on a project: How many 5-person teams can be chosen? C(12,5) If two people insist on working together (or not working at

all), how many 5-person teams can be chosen? C(10,3)+C(10,5)

If two people insist on not working together, how many 5-person teams can be chosen? C(10,5)+C(10,4)*2

How many 5-person teams consist of 3 men and 2 women?C(5,3)*C(7,2)

How many 5-person teams contain at least 1 man? C(12,5)-C(7,5) How many 5-person teams contain at most 1 man?

C(7,5)+C(5,1)*C(7,4)

Similar Exercises

An instructor gives an exam with 14 questions. Students are allowed to choose any 10 of them to answer: Suppose 6 questions require proof and 8 do not:

How many groups of 10 questions contain 4 that require a proof and 6 that do not?

How many groups of 10 questions contain at least one that require a proof?

How many groups of 10 questions contain at most 3 that require a proof?

A student council consists of 3 freshmen, 4 sophomores, 3 juniors and 5 seniors. How many committees of eight members contain at least one member from each class?

Permutations with Repetition

Suppose a collection consists of n objects of which n1 are of type 1 and are indistinguishable from each other

n2 are of type 2 and are indistinguishable from each other…

nk are of type k and are indistinguishable from each other,

and suppose n1+n2+ …+nk =n. Then the number of distinct permutations of the n objects is

Example: How many distinguishable orderings of the letters in the word MISSISSIPPI are there?

!!!

!

21 knnn

n

Combinations with Repetition

An r-combination with repetition allowed is an unordered selection of elements where some elements can be repeated.

The number of r-combinations with repetition allowed from a set of n elements is C(r + n –1, r)

Example: How many triples of integers from 1 through n can be formed in which the elements of the triple are in increasing order but not necessarily distinct?

Integral Equations

How many non-negative integer solutions are there to the equation x1 + x2 + x3 + x4 = 10?

How many positive integer solutions are there for the above equation?

Algebra of Combinations and Pascal’s Triangle

The number of r-combinations from a set of n elements equals the number of (n – r)-combinations from the same set. Namely, C(n,r)=C(n, n-r) Show that C(n, 0)2 + C(n, 1)2 + … + C(n, n)2 =

C(2n, n). Pascal’s Formula:

Let n and r be positive integers and r ≤ n. Then C(n + 1, r) = C(n, r – 1) + C(n, r)

Binomial Theorem

(a + b)n = C(n, k)an-kbk

Show that C(n, k) = 2n

Show that (-1)kC(n, k) = 0. Express C(n, k)3k in the closed form.