Fundamental Counting Principles:
Let E1 and E2 be two events. The first event E1 can occur in m different ways. After E1 has occurred, E2 can occur in n different ways. The number of ways that the two events can occur is
m * n
The Fundamental Counting Principle can be extended to three or more events.
EXAMPLE
1. How many different trios of letters from the English alphabet are possible if the first letter cannot be Z?
(Possible first letters) * (Possible second letters) * (Possible third letters)
(25) * (26) * (26)
16,900
PERMUTATIONS
Definition: a permutation of n different elements is an ordering of the elements such that one element is first, one is second, one is third, and so on.
EXAMPLE
1.How many permutations are possible for the letters A, B, C, D?
First Position = any of the 4 letters
Second Position = any of the remaining 3 letters
Third Position = any of the remaining 2 letters
Fourth Position = the one remaining letter
Total number of permutations = 4 * 3 * 2 * 1 = 4! = 24
Number of Permutations of n Elements:
the number of permutations of n elements is
n * (n-1) ...4 * 3 * 2 * 1 = n!
In other words, there are n! different ways that n elements can be ordered.
Permutations of n Elements Taken r at Time:
the number of permutations taken r at a time is
EXAMPLE
1. There are 10 horses in a race. In how many ways can there be 1st, 2nd, 3rd, 4th, and 5th place?
10! / (10-5)! = 10*9*8*7*6 = 30,240
COMBINATIONS
Definition: permutations are when order does matter, but combination simply counts the number of permutations while order does NOT matter.
Combinations of n Elements Taken r at a Time:
the number of combinations of n elements taken r at a time is
EXAMPLE
1. You have a standard deck of 52 cards and will deal them out in sets of 5. How many different hands are possible?
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