Wednesday, February 28, 2007

Chapter 9 Section 5: Probabilities of Various Combinations

Section 5 is about Permutations and Combinations. In a given situation, it is important that you realize whether the situation is one where the order of elements matters or does not matter. This can be explained by the definitions of permutations and combinations.

Important Definitions:

Combination: A subset of elements where the order of the elements does not matter.

Computation of Number of Combinations:

Number of combinations = Total number of permutations / Number of permutations of each one combination

nCr = number of different combinations of n elements taken r at a time.

n = how many elements in a group.
r = how many elements we are selecting.

Remember that last section we learned about Permutations.
A permutation of a set of objects is an arrangement in a definite order of some or all of the elements in that set. So basically, order matters with permutations.

nPr = number of different permutations of n elements taken r at a time.

n = how many elements in a group.
r = how many elements we are selecting.

(To access these keys on your calculator: MATH -> PRB -> nPr or nCr )

When I was talking about the order mattering versus the order not mattering I meant that, for example, if you have the numbers 1, 2, 3, and 4 and you are making three digit numbers using these four numbers, in a permutation, order matters. So, if you said 123 and also 321 that would count as two separate numbers that you could make. With a combination, order does not matter, so when you use the numbers 123 that is the only time you will count it as a number. If you decided to put them in the order 321, that would not count as a separate number that you made. When the numbers 1, 2, and 3 are used in the same number for a combination, you do not count the different ways that the numbers can be arranged as separate numbers. So, with the numbers 1, 2, 3, 4, there are 24 different permutations, but only 4 different combinations.
To prove this, put in your calculator:

For the permutation:
4 nPr 3 and you should get 24.

For the combination:
4 nCr 3 and you should get 4.

Calculation of Number of Permutations of Combinations:

nPr = (n!)/((n-r)!)and nCr = (n!)/(r!(n-r)!)

Remember: ! is the symbol for a factorial.
For example (8!) = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 40,320


Example Problem:

#1.) 15 C 5 = ?

This is a combination, so nCr = (n!)/(r!(n-r)!)

15C5 = (15! )/( 5! (15-5)! )
15C 5 = (15!) / (5! 10! )
15 C 5 = 3003

Note that the numerator is the same number as the number that precedes the "C." The 5 and the 10 (the numbers in the denominator) add up to the numeral in the numerator.

To solve this on the calculator, you would type in: 15 nCr 5 which equals 3003.

#2.) You have 15 CD's that you absolutely love, but, sadly, you only have room for 5 of them in your car. In how many different ways could the group of 5 CD's be chosen?

The word "group" tells you that a number of combinations is being asked for.
You can solve this problem two ways, either manually, or on your calculator. It is useful to know how to solve both ways, because we have both calculator and non-calculator portions to our tests and quizzes.

On the calculator:

15 nCr 5 = 3003

So, there are 3003 different ways to choose the group of 5 CD's.

By hand:

nCr = (n!)/(r!(n-r)!)

15 C 5 = (15!)/ (5! (15-5)!)

15 C 5 = 3003


Link:

http://www.saliu.com/oddslotto.html

This site is to help calculate lotto odds, but it is actually a good explanation of combinations.



This is a photo of a bay in Greece at night. I used this photo as the basis for a painting.



"Poppies at Argenteuil", Claude Monet 1873
I started to paint this picture after Monet awhile ago but I have been so busy that I have had no time to work on it. The unfinished painting is hanging in my living room. Hopefully I can get some time to finish...maybe after SATs.


Tara! You’re next!



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