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Fundamental Counting Principle PERMUTATIONS AND COMBINATIONS
Fundamental Counting Principle
If there are n(A) ways in which an event A can occur, and if there are n(B) ways in which a second event B can occur after the first event has occurred, then the two events can occur in n(A) · n(B) ways.
Special products such as 4! (4 · 3 · 2 · 1) (or any other constant) frequently occur in counting theory. This symbol is a special notation, known as factorial. Factorial is explained as:
For any positive integer n, we define n-factorial, written as n! = n(n - 1)(n - 2)(n - 3)………..
We define 0! =1
Permutation and Combination Formulas
Permutation - The number of possible distinct arrangements of r objects chosen from a set of n objects is called the number of permutations of n objects taken r at a time and it equals:
nPr = __n!__ (n – r)!
Permutation and Combination Formulas
Example In how many ways can a president, vice president, secretary, and treasurer be selected from an organization with 20 members?
Solution (the number of arrangements in which 4 people can be selected from a group of 20) n = 20 r = 4
nPr = 20!__ = 20 · 19 · 18 · 17 · 16! = 116,280
(20 - 4)! 16!
Permutation and Combination Formulas
Combination - The number of combinations of n objects taken r at a time is:
nCr = ___n!___ r!(n – r)!
Permutation and Combination Formulas
Example In the Texas lottery you choose 6 numbers from 1 though 54. If there is no replacement or repetition of numbers, how many different combinations can you make?
Solution n = 54 r = 6
nCr = 54!__ = 54 · 53 · 52 · 51 · 50 · 49 = 25,827,165
6! (54-6)! 720
Fundamental Counting Principle
If there are n(A) ways in which an event A can occur, and if there are n(B) ways in which a second event B can occur after the first event has occurred, then the two events can occur in n(A) · n(B) ways.
Special products such as 4! (4 · 3 · 2 · 1) (or any other constant) frequently occur in counting theory. This symbol is a special notation, known as factorial. Factorial is explained as:
For any positive integer n, we define n-factorial, written as n! = n(n - 1)(n - 2)(n - 3)………..
We define 0! =1
Permutation and Combination Formulas
Permutation - The number of possible distinct arrangements of r objects chosen from a set of n objects is called the number of permutations of n objects taken r at a time and it equals:
nPr = __n!__ (n – r)!
Permutation and Combination Formulas
Example In how many ways can a president, vice president, secretary, and treasurer be selected from an organization with 20 members?
Solution (the number of arrangements in which 4 people can be selected from a group of 20) n = 20 r = 4
nPr = 20!__ = 20 · 19 · 18 · 17 · 16! = 116,280
(20 - 4)! 16!
Permutation and Combination Formulas
Combination - The number of combinations of n objects taken r at a time is:
nCr = ___n!___ r!(n – r)!
Permutation and Combination Formulas
Example In the Texas lottery you choose 6 numbers from 1 though 54. If there is no replacement or repetition of numbers, how many different combinations can you make?
Solution n = 54 r = 6
nCr = 54!__ = 54 · 53 · 52 · 51 · 50 · 49 = 25,827,165
6! (54-6)! 720
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