Permutations and Combinations April 11th

Hey, today we started off with Mental Math, which was really hard.

Also Mr.Max before I get started, because your such a vocabulary expert, take a stab at his question,

Is "sidewards" a word?....(don't try to look it up until you think you know the answer.)

Anyways, We then went over perms and coms. We learned and reviewed the formula for permutations that nPr=n!/(n-r)! and the formula for combinations it nCr=n!/(n-r)!*r!

Lets use one of the formulas in a question and see how it works...( we did this one in class )

Ok, you have 3 & 4 digit codes with 4 digits to choose from. (1,2,3,4)

Since you're trying to find out possible codes, it matters what order the numbers are in.

And since it matters what order the numbers are in, it is therefore a permutation. The formula for permutations is nPr=n!/(n-r)! where (n) is the total number of objects to be ordered or considered and (r) is how many objects at a time are ordered. ( If you have any other questions about factorials I put up a post under factorials that you can read, it might be of some help, Mr.Max also said that these formulas would be a good thing to put on a formula sheet.)

Anyways, so this works out to 4P3=4!/1!,

4! is the same as 4*3*2*1=24,

1! is the same as 1=1

therefore 4P3= 24/1

there are 24 ways to get 3 digit codes using the digits 1,2,3, and 4.

Then you had to figure out how many 4 digit codes you could have with the numbers 1,2,3 and 4. You still use the same formula.

Except this time, it works out to:

4P4=4!/0!,

4! is the same as 24

0! is by definition equal to 1

this leave us with 4P4= 24/1 which is 24

This means that there are 24 ways to get a 3 digit code and 24 ways to get a 4 digit code.

A different way of getting the same answer is with your calculator. From your home screen you hit (n) (whatever the case may be) then MATH, over to PRB, then down to nPr or nCr, (what ever the case may be), and then ENTER, and then put in (r) (whatever the case may be)

Mr.Max said that you don't need to use these formulas if you don't want to, but you should understand then.

Also Mr.Max will be gone Tuesday, Wednesday, Thursday, and Friday, of next next week, ( I think that would be the 24th, 25th, 26th, and 27th, and he has the joy of going to talk to a bunch of teachers or something. He didn't look to enthusiastic about that. We will have a sub and probably just work on Accelerated Math, and other assigned questions.

Practice questions due for tomorrow are from the handout, #5, #11, and #14

This is a link that might be of some use for you.

http://www.themathpage.com/aPreCalc/permutations-combinations.htm

Combinations

I really like the previous posts because they explain what i cannot. So will do what I can to add to it.





We can also use pascals triangle to figure out combinations, the triangle has infinite amount of rows and can go on forever. If, for instance, we are trying to find 10_C_4, we would take the fourth entry in the tenth row which would come out to 210, so this means that there are 210 ways to choose 4 numbers out of a set of 10 numbers.

This can be done from either side and you should get the same answer.

Sometimes My Light Bulb Goes 'ON'....

I am so impressed with these last few posts. You (who have posted so far) have taken some fairly big risks, since to my knowledge, there's no right way to do this....and you're doing it really well!

At risk of gushing, I'm proud of how hard you guys as a class are trying to make these little brainwaves of mine work. This is an example of why we're blogging and why I truly believe that technologies such as our blog can make the whole greater than the sum of its parts.

Really neat stuff folks, and as your teacher, I'm glad to see you learn on purpose.

RM

My Take On Permutations

First off I would like to say that this is a interesting way to learn our new unit, but it will also be quite interesting to see what we all come up with and if well we know what were talking about after all. So here i go;;;;

A permutation is an arrangement of objects in different orders. The order of the arrangement is important!!
For example, the number of different ways 3 students can enter school can be shown as 3!, or 3·2·1, or 6. There are six different arrangements, or permutations, of the three students in which all three of them enter school.

The notation for a permutation: nPr
n is the total number of objects
r is the number of objects chosen (want)
(Note if n = r then nPr = n!)

Some examples of Permutations:
1. 5P5 = 5·4·3·2·1 = 120
2. 7P5 = 7·6·5·4·3 = 2520

Here are some Questions you may see and how you can figure out how to do them using the Permutations Formula (nPr) in use!!

1. What is the total number of possible 5-letter arrangements of the letters s,w,i,n,g if each letter is used only once in each arrangement?
5P5 = 5·4·3·2·1 = 120
( 5 letters to choose from, n#, want 5-letter arrangements, r#)

2. How many different 3-digit numerals can be made from the digits of 45678 if a digit can appear just once in a numeral?
5P3 = 5·4·3 = 60
( 5 numbers to choose from, n#, want 3-digit numerals, r#)

Also, as I was searching around I found some interesting websites on Perms & Cons that were helpful to me and thought I would share them with the rest of the class;

1. http://mathforum.org/dr.math/faq/faq.comb.perm.html ( This website gives the break down of how you start out on a question and take the information and put it into the formula)

2. http://regentsprep.org/regents/math/permut/PracPerm.htm (If your not to sure your doing this unit to right you can try these practice questions with the answers)

Combinations -Revised

Here is everything Wikipedia has to say about mathematical combinations:

http://en.wikipedia.org/wiki/Combinations

to summarize;

Combination: a combination is an unordered collection of unique elements
ie: numbers, variables

- the order of the combination is not important

- two lists with the same elements in a different order are considered to be the same combination

- the elements cannot be repeated, they must appear once, thus making them "unique"

- combinations are defined by the elements contained in them

http://www.dictionary.com/ states that a combination is:
the arrangement of elements into various groups without regard to their order in the group.

This is a link I found on http://www.reference.com/ to
Many Common types of permutation and combination math problems, with detailed solutions
I'm not sure how useful it will be, I browsed through and it seemed to have potential, so hopefully it provides some useful information.

This link leads to some random school districts exam prep page, it looks very very useful, it has lessons and practice and formulas, yay!
http://regentsprep.org/regents/math/math-topic.cfm?TopicCode=combin