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Factorial Notation The expression 6 × 5 × 4 × 3 × 2 × 1 = can be written as 6!, which is read as “six factorial.”
In general, n! is the product of all the counting numbers beginning with n and counting backwards to 1.
We define 0! to be 1. Factorial on your TI calculator.
Example :Find the value of each expression:a) 3!
b) 0!
c) 3! + 2!
d)
Fundamental Counting Principle:If one activity can occur in any of m ways and, following this, a second activity can occur in any of n ways, then both activities can occur in the order given in m*n ways.
Permutation Formula an arrangement of objects in some specific orderIn general P(n, r) means the number of permutations of n items arranged r at a time.
The formula for permutation is
Permutations on your TI calculator
Note:nPn = n!
3P3 = 3! = 3*2*1 = 6
Words used in permutation problems:• arrangement• line up• president, vice president, secretary• 1st, 2nd, 3rd place
Example :A license plate begins with three letters. If the possible letters are A, B, C, D and E, how many different arrangements of these letters can be made if no letter is used more than once?
Permutations with repetitionIf we want to arrange items when there are more than one of the same item, we need to divide by the number of identical items:Example:Find the number of arrangements of the letters that can be formed from the letters IDENTITY, using each letter
Solution:Example: Find the number of arrangements of letters that can be formed from the letters:
MINIMUMSolution:
CombinationsAn arrangement of objects in which the order is not important is called a combination. This is different from permutation where the order matters. For example, suppose we are arranging the letters A, B and C. In a permutation, the arrangement ABC and ACB are different. But, in a combination, the arrangements ABC and ACB are the same because the order is not important.
The number of combinations of n things taken r at a time is written as C(n, r).
The formula is given by:
Combinations on your TI calculator
Example: In how many ways can a coach choose three swimmers from among five swimmers?
Words used in combination problems:• committee• group• team
Example 6:There are 5 red and 4 white marbles in an urn. A marble is drawn from the urn and not replaced. Then, a second marble is drawn. a. In how many ways can a red marble and a white marble be drawn in that order?
b. In how many ways can a red marble and a white marble be drawn in either order?
Example 7:An urn contains three white balls and four red balls. Two balls are chosen at random. How many ways can you chose at least one of the red balls?