Combinations (11-7)

Combinations

Vocabulary

Combination- a grouping of objects or events in which the order does not matter.

Examples

The number of k-combinations for all k is the number of subsets of a set of n elements. There are several ways to see that this number is 2ⁿ. In terms of combinations, , which is the sum of the nth row (counting from 0) of the binomial coefficients in Pascal's triangle.

Fundamental Principle of Counting:  (also known as the multiplication rule for counting)  If a task can be performed in n₁ ways, and for each of these a second task can be performed in n₂ ways, and for each of the latter a third task can be performed in n₃ ways, ..., and for each of the latter a k^th task can be performed in n_k ways, then the entire sequence of k tasks can be performed in  n₁ • n₂ • n₃ • ... • n_k  ways.

Permutation:  A set of objects in which position (or order) is important. To a permutation, the trio of Brittany, Alan and Greg is DIFFERENT from Greg, Brittany and Alan.  Permutations are persnickety (picky). Combination:  A set of objects in which position (or order) is NOT important. To a combination, the trio of Brittany, Alan and Greg is THE SAME AS Greg, Brittany and Alan.                         Let's look at which is which: Permutation       versus       Combination 1. Picking a team captain, pitcher, and shortstop from a group. 1. Picking three team members from a group. 2.  Picking your favorite two colors, in order, from a color brochure. 2.  Picking two colors from a color brochure. 3.  Picking first, second and third place winners. 3.  Picking three winners.

Formulas:

A permutation is the choice of r things from a set of n things without replacement and where the order matters.
SpecialCases:
A combination is the choice of r things from a set of n things without replacement and where order does not matter.  (Notice the two forms of notation.)
Special Cases:

  

The term "combination" lock is mathematically confusing.  To open such a lock, the "order" of the digits entered IS very important, unlike a mathematical combination. New Name:  Permutation Lock

Example 1:   
Evaluate  :  
                      

Notice how the cancellation occurs, leaving only 2 of the factorial terms  in the numerator.  A pattern is emerging ... when finding a combination  such as the one seen in this problem, the second value (2) will tell you  how many of the factorial terms to use in the numerator, and the  denominator will simply be the factorial of the second value (2).

Example 2:   Joleen is on a shopping spree.  She buys six tops, three shorts and 4 pairs of sandals.  How many different outfits consisting of a top, shorts and sandals can she create from her new purchases?

(6)(3)(4) = 72 possible outfits
 

Example 3:  What is the total number of possible 4-letter arrangements of the letters
  m, a, t, h,  if each letter is used only once in each arrangement? 
              or      or   simply 4!

Example 4:  
There are 12 boys and 14 girls in Mrs. Schultzkie's math class.  Find the number of ways Mrs. Schultzkie can select a team of  3 students from the class to work on a group project.  The team is to consist of 1 girl and 2 boys.     

Order or position, is not important.  Using the multiplication counting principle,

 http://www.regentsprep.org/regents/math/algtrig/ats5/lcomb.htm

Video Help

https://www.youtube.com/watch?v=bCxMhncR7PU

http://study.com/academy/lesson/what-is-a-mathematical-combination.html

http://www.shmoop.com/video/combinations