How many distinct permutations of the letters of the word ALFALFA are there?
Would this be 210?
yes
To find the number of distinct permutations of the word ALFALFA, we need to determine the total number of arrangements of the letters.
The word ALFALFA has 7 letters, including 3 A's and 2 L's.
To find the number of distinct permutations, we can use the formula for permutations with repeated elements.
The formula is given by:
n! / (n1! * n2! * ... * nk!), where:
n is the total number of elements,
n1, n2, ... , nk are the counts of the repeated elements.
In this case, n = 7, n1 = 3 (A's), and n2 = 2 (L's).
So the calculation will be:
7! / (3! * 2!)
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
3! = 3 * 2 * 1 = 6
2! = 2 * 1 = 2
Substituting these values into the formula:
5040 / (6 * 2) = 5040 / 12 = 420
Therefore, there are 420 distinct permutations of the letters in the word ALFALFA, not 210.
To calculate the number of distinct permutations of the letters in the word "ALFALFA," we can use the formula for permutations with repeated elements.
The formula for permutations with repeated elements is n!/(n1! * n2! * ... * nk!), where n is the total number of elements, and n1, n2,..., nk represent the number of times each element is repeated.
In the word "ALFALFA," we have a total of 7 letters, with "A" repeated 4 times, "L" repeated 2 times, and "F" repeated 1 time.
Using the formula:
Number of distinct permutations = 7! / (4! * 2! * 1!)
Calculating this:
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
4! = 4 * 3 * 2 * 1 = 24
2! = 2 * 1 = 2
1! = 1
Substituting these values:
Number of distinct permutations = 5040 / (24 * 2 * 1) = 5040 / 48 = 105
Therefore, there are 105 distinct permutations of the letters in the word "ALFALFA."