Sequences

Sequences: An ordered list of numbers that follow a certiain pattern (or rule).

Example:

4, 7, 10, 13, 16, 19, 22

Use the following process to help find a formula for the pattern.

n: is the position or place value
tn: value of that position

When looking at the pattern this way, it is easy to see the it just adding three. So the formula would be.

tn= n-1+3


Or

tn=7n-9



Arithmetic Sequences:

(i) Recursive defintion: An order list of numbers generated by continnous adding value (the common)

(ii) Implictit defintin: An oredered list of numbers where each number n the list is generated by a linear equation.

Common Difference (d):

(i) The number that is repeatedly added to successive term in an arithemic sequence.

(ii) Fommr the implicit definitiion, d is the slope of the linear equation. 

Recursivw: add to what you have. Ex: tn= tn-1+3

Explicit: a formula that uses the position or "n" value to find. Ex: tn= 3n+7

You should also know how to find the common difference

        d = tn - t(n - 1)

        d is the common difference

        tn is an arbitrary term in the sequence

        t(n - 1) is the term immediately before tn in the sequence

As well you should know how to find the nth term in an arithmetic sequence

        tn = a + (n - 1)d

        tn is the nth term

        a is the first term

        n is the "rank" of the nth term in the sequence

        d is the common difference

Perms and Combs, Videos attached

Permutations and combinations are often misused and interchanged comfortably, along with arrangements, collections and groupings. It’s often easy to misuse them, but there is a slight difference in their meanings:

Permutations: An arrangement of a set of events or objects, where in the order of the events MATTERS.

Example: The arrangement of the numbers 1, 2, 3 and 1, 3, 2 is a different PERMUTAION.

Combinations: An arrangement of a set of events or objects, where in the order DOES NOT MATTER.

Example (follow-up): The arrangement of the numbers 1, 2, 3 and 1, 3, 2 is THE SAME COMBINATION.

The knowledge of the difference of meaning in these concepts is crucial.

There are different mathematical formulas for calculating permutation and combination arrangements. It is very important, however, to be able to logically analyze if the order in the arrangement matters.

For permutations:

 For combinations:

The only difference between the two is that in the combination formula, it is essential to divide the result by the number of ways the objects or events can inter-arrange to switch their order, because the order is irrelevant.

The following videos provide a very helpful visual guide to help further develop an understanding of these concepts:

http://www.khanacademy.org/math/probability/v/permutations-and-combinations-1

http://www.khanacademy.org/math/probability/v/permutations-and-combinations-2

http://www.khanacademy.org/math/probability/v/permutations-and-combinations-3

http://www.khanacademy.org/math/probability/v/permutations-and-combinations-4

perms and combs

Permutations, has several different meanings, but all are related to “the act of permuting”, rearranging objects and values. They occur in almost every domain of mathematics. It is an arrangement of numbers in an order . For example there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1).

The formula for permutation is:

                                nPr =n!/(n-r)

Combination- is a way of selecting several things out of a large group where the order doesn’t matter. In smaller cases you can count the number of combination. When the set gets larger you have to use difficult mathematics to find the number of combinations.

The combination formula:

              (nk)=n(n-1)…(n-k+1)/k(k-1)…1

Premutations and Combinations

Permutations- A permutation is a order of numbers where the order does matter
Combinations- A Combination is a order of numbers where the order does not matter

A combination lock is a perfect example of a Permutation as it is three order number line where the numbers have to be in the right order to open the lock.

If your combo is 23-67-9 you can only put it in as this order if you put 67-23-9 the numbers are the same but the order is not making it impossible to open your locker.

The order through which you put fruit into a smoothie is a combination
If Apples are 1
Banannas are 2
Peaches are 3
The order can go many ways
123
132
321
213
231
312

In other words a permutation is an ordered combination

The above formula is for Permutations without repitition the below fomule is for combinations without repitition.

where n is the number of things to choose from, and you choose r of them       (Order does not matter)

For things where repitition is allowed the formulas are as follows

Permutations is simply N to the exponent R where N is the number of things to choose from and R is how many you will choose

Combinations where repitition is allowed is shown as the formula below

where n is the number of things to choose from, and you choose r of them
(Repetition allowed, order doesn't matter)

All information was taken from the link below the above descriptions and formulas are a short sumamry of the site the site as well provides real world examples

http://www.mathsisfun.com/combinatorics/combinations-permutations.html

Combinations & Permutations.

Combinations are easy going. Order doesn’t matter. You can mix it up and it looks the same. Let’s say I’m a cheapskate and can’t afford separate Gold, Silver and Bronze medals. In fact, I can only afford empty tin cans.
How many ways can I give 3 tin cans to 8 people?
Well, in this case, the order we pick people doesn’t matter. If I give a can to Alice, Bob and then Charlie, it’s the same as giving to Charlie, Alice and then Bob. Either way, they’re going to be equally disappointed.
This raises an interesting point — we’ve got some redundancies here. Alice Bob Charlie = Charlie Bob Alice. For a moment, let’s just figure out how many ways we can rearrange 3 people.
Well, we have 3 choices for the first person, 2 for the second, and only 1 for the last. So we have 3 * 2 * 1 ways to re-arrange 3 people.
Wait a minute… this is looking a bit like a permutation! You tricked me!
Indeed I did. If you have N people and you want to know how many arrangements there are for all of them, it’s just N factorial or N!
So, if we have 3 tin cans to give away, there are 3! or 6 variations for every choice we pick. If we want to figure out how many combinations we have, we just create all the permutations and divide by all the redundancies. In our case, we get 336 permutations (from above), and we divide by the 6 redundancies for each permutation and get 336/6 = 56.
The general formula is

which means “Find all the ways to pick k people from n, and divide by the k! variants”. Writing this out, we get our combination formula, or the number of ways to combine k items from a set of n:

http://www.mathsisfun.com/combinatorics/combinations-permutations-calculator.html

Enjoy!

Permutations

Permutations - An arrangement of 'n' (number) objects taking all or some of the objects at a time where the order does matter.

Without Repetition:

            Formula:  nPr = n!/(n-r)!

n - The total number of items to choose from.
r - The number of items being chosen.

ex. a) How many ways can you order billiard balls numbered 1 to 16?
 b) How many ways can you order 5 different balls?


a)
An easy way to do it is insert [1],[6],[MATH], go over to the 'Probability' option and select option 4 so it will look like this on your calculator:

16!

Really all '!' is, is a short cut for going 16x15x14x13x12xx11x10x9x8x7x6x5x4x3x2x1
answer: 20922789888000 or 2.09x10¹³

b)
 You can use the formula nPr = n!/(n-r)! 
n=16
r=5                     =16!/(16-5)!
                           =16!/11!
                           =524160
or with your calculator you can take a shortcut by inserting "16" then [MATH] button, over to probability and select option 2 "nPr"than insert the number "5". It should look like this on your calculator.

16 nPr 5

Hit solve and you should get the same answer as above: "524160".