Wednesday, May 13, 2009

Combinatorics BOB!

This unit was all on Combinatorics. We were exposed to:
Fundamental Principle of Counting, Factorial Notation, Permutations, Permutations of Non-distinguishable objects, Circular permutations, Combinations, and Binomial Theorem. (so much for a short unit, huh Mr.K? XD)

(8) Break it down!

Fundamental Principle of Counting:
When you have Q ways to go a first thing, and P ways to do a second thing, than you'd have QP ways of doing both things. Genius!

Factorial Notation: When you multiply consecutive numbers. example: n! = (n)(n-1)(n-2)(n-3)..etc. To solve a Factorial Notation, 3! for example, you can multiply (3)(2)(1) or punch in 3, [MATH], left arrow key, .

Permutations:
Ordered arrangement of objects, without repetition. Its when you "pick", meaning order matters. nPr = (n!)/(n-r)!. where n = # of objects to pick from, r = # of objects to be arranged in, and read as n PICK r. [MATH], left arrow key, .
• example

Permutations of Non-distinguishable objects:
When you arrange objects n ways, where some of its objects are non-distinguishable. This means you can't tell the objects apart from each other, and it wouldn't really matter which one of those are which. In cases like this, you'd have repetition of the same result. To get rid of these repetition, you would divide by them. (n!)/(k1!k2!k3). The the numbers beside the K's are subscript.
• example:

Circular permutations: When you have # of ordered arrangements that can be made of n objects in a circle. (n-1)!. Since its in a circle, there is no end or start. So she'd have to make one.
• SPECIAL CASE!: some circles can be flipped over, so the # of arrangements would be divided by two. (so no repetitions). [(n-1)!]/2

Combinations: When the order of chosen objects does not matter. The CHOOSE formula: (n/r) or nCr = (n!)/[(n-r)!(r!)], where n = # objects to choose from, r = # objects to be arranged.[MATH], left arrow key, .

Binomial Theorem: There's a super super long formula for this...and i refuse to type out. You can find it in May 11 slides, 15/17. Its what helps you figure out binomials, (a+b)^n, but remember the patterns and you're set.
• The coefficients in each term can be found by Pascal's Triangle, nC(i-1).
• b's exponent always starts out as b^0 in the first term, and b^n last term.
• n is the sum of the exponent's on a and b.
• The number of terms is n+1, because

k im taking a breather guys, this took longer than expected. XD
examples will be scanned

so much yellow....

Overall, this unit was tough. I would always have trouble telling if the question was a permutation, or combination, etc. and then you have questions like [THESE], who fall into more than one category. D:
Hopefully we'll work on more questions tomorrow in class, and have things straighten out.

good luck!
mary~