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 kth task can be performed in nk ways, then the entire sequence of k tasks can be performed in n1 • n2 • n3 • ... • nk 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. /Special Cases:
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 :
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 of1 girl and2boys.
Order, or position, is not important. Using the multiplication counting principle,