Number Blunder

1. Factorial Notation

Factorial notation is used to write the product of all the positive whole numbers up to a given number.

Definition of n!

n factorial is the product of all the integers from 1 to n

"n factorial" is written with an exclamation mark as follows: n!

n! = (n)(n − 1)(n − 2)...

Example:

5! = 5 × 4 × 3 × 2 × 1 = 120

2. Counting

Number of Outcomes of an Event

E = an event

n(E) = number of outcomes of event E

Example:

E = "hours per day"

n(E)=24

So there are 24 hours per day

Let E1 and E2 be mutually exclusive events.

Let event E describe the situation where either event E1 or event E2 will occur.

The number of times event E will occur can be given by the expression:

n(E) = n(E1) + n(E2)

Example:

In how many ways can a number be chosen from 1 to 22 such that it is a multiple of

3 or 8?

Here, E1 = multiples of 3:

E1 = {3, 6, 9,12, 15, 18, 21}

n(E1) = 7

E2 = multiples of 8:

E2 = {8, 16}

n(E2) = 2

Events E1 and E2 are mutually exclusive.

n(E) = n(E1) + n(E2) = 7 + 2 = 9

Multiplication Rule

Now consider the case when two events E1 and E2 are to be performed and the events E1 and E2 are independent events

Example:

For our clothes problem above, say we found 3 caps that we could wear with our 2 t-shirts and 4 pairs of jeans. How many different combinations could we choose from now?

We have 2 choices in the first row, 4 in the second row and 3 in the third row. Together, we will have n(E) = n(E1) × n(E2) = 2 × 4 × 3 = 24 combinations

3. Permutations

An arrangement or ordering of a set of objects is called a permutation.

In a permutation, the order that we arrange the objects in is important.

Theorem 1 - Arranging n Objects

In general, n distinct objects can be arranged in n! ways.

Example:

In how many ways can 4 different resistors be arranged in series?

Since there are 4 objects, the number of ways is

4!=24 ways

Theorem 2 - Number of Permutations

The number of permutations of n distinct objects taken r at a time, denoted by Pnr, where repetitions are not allowed, is given by Pnr=n(n−1)(n−2)...(n−r+1)=n! ÷ (n−r)!

Example:

In how many ways can a supermarket manager display 5 brands of cereals in 3 spaces on a shelf?

This is asking for the number of permutations, since we don't want repetitions. The number of ways is:

P53=5! ÷ (5−3)! =5! ÷ 2! =60

Theorem 3 - Permutations of Different Kinds of Objects

The number of different permutations of n objects of which n1 are of one kind, n2 are of a second kind, ... nkare of a k-th kind is n! ÷ (n1!×n2!×n3×...×nk!)

Example:

In how many ways can the six letters of the word "mammal" be arranged in a row?

Since there are 3 "m"s and 2 "a"s in the word "mammal", we have:

6! ÷ 3!2!=60

There is one "L" in "mammal", but it does not affect the answer, since 1! = 1.

Theorem 4 - Arranging Objects in a Circle

There are (n−1)! ways to arrange n distinct objects in a circle.

Example:

In how many ways can 5 people be arranged in a circle?

(5−1)!=4!=24 ways

4. Combinations

A combination of n objects taken r at a time is a selection which does not take into account the arrangement of the objects. That is, the order is not important.

Number of Combinations

The number of ways (or combinations) in which r objects can be selected from a set of n objects, where repetition is not allowed, is denoted by:

Cnr=n! ÷ r!(n−r)!

Example:

Find the number of ways in which 3 components can be selected from a batch of 20 different components.

C203 =20! ÷ 3!(20−3)! =20! ÷ 3!17! =1140

5. Probability

Definition of a Probability

Suppose an event E can happen in r ways out of a total of n possible equally likely ways.

Then the probability of occurrence of the event is denoted by P(E)=r ÷ n

The probability of non-occurrence of the event is denoted by P(E⎯)=(n−r)÷n=(1−r)÷n

Notice the bar above the E, indicating the event does not occur.

Thus, P(E⎯)+P(E)=1

In words, this means that the sum of the probabilities in any experiment is 1.

Definition of Probability using Sample Spaces

When an experiment is performed, a sample space of all possible outcomes is set up.

In a sample of N equally likely outcomes, a chance of 1 ÷ N is assigned to each outcome.

The probability of an event for such a sample is the number of outcomes favorable to E divided by the total number of equally likely outcomes in the sample space S of the experiment.

P(E)=n(E) ÷ n(S)

where:

n(E) is the number of outcomes favorable to E

n(S) is the total number of equally likely outcomes in the sample space S of the experiment.

Properties of Probability

(a) 0 ≤ P(event) ≤ 1

In words, the probability of an event must be a number between 0 and 1.

(b) P(impossible event) = 0

In words: The probability of an impossible event is 0.

(c) P(certain event) = 1

In words: The probability of an absolutely certain event is 1.

Example:

What is the probability of...

(a) Getting an ace if I choose a card at random from a standard pack of 52 playing cards.

There are 4 aces in a normal pack. So the probability of getting an ace is:

P(ace)=4/52=1/13

(b) Getting a 5 if I roll a die.

A die has 6 numbers.

There is only one 5 on a die, so the probability of getting a 5 is given by:

P(5)=1/6

(c) Getting an even number if I roll a die.

Even numbers are 2,4,6. So

P(even)=3/6=1/2

(d) Having one Tuesday in this week?

Each week has a Tuesday, so probability = 1.