Find the number of distinguishable permutations of the word EDUCATED.
8 letters with 2 E's, 2 D's
number of permutations = 8!/(2!2!) = 10080
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To find the number of distinguishable permutations of the word EDUCATED, we need to consider the total number of letters and account for any repeating letters.
Step 1: Determine the total number of letters.
The word EDUCATED has a total of 8 letters.
Step 2: Identify any repeating letters.
In the word EDUCATED, the letters E and D are repeated. The letter E appears twice, and the letter D appears twice as well.
Step 3: Calculate the number of distinguishable permutations.
To calculate the number of distinguishable permutations, we use the formula: n! / (n1! * n2! * ... * nk!)
In this formula:
- n is the total number of letters.
- n1, n2, ..., nk are the repetitions of each letter.
Using this formula, we can calculate the number of distinguishable permutations as follows:
Number of permutations = 8! / (2! * 2!)
= (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (2 * 1 * 2 * 1)
= 40,320 / 4
= 10,080
Therefore, there are 10,080 distinguishable permutations of the word EDUCATED.
To find the number of distinguishable permutations of the word "EDUCATED," we first need to calculate the total number of arrangements considering all the letters.
The word "EDUCATED" contains 8 letters, but there are repeated letters. So, we need to adjust for these repetitions to count only the unique permutations.
1. Count the number of times each letter appears in the word "EDUCATED."
- E appears twice,
- D appears twice,
- U, C, and A appear once each,
- T appears once.
2. Calculate the total number of arrangements by considering the repeated letters. We can use the formula for permutations of a multiset.
Number of arrangements = n! / (n1! * n2! * n3! * ... * nk!)
- n is the total number of letters (8 in this case).
- n1, n2, n3, ... nk represents the count of each repeated letter.
Substituting for the values:
Number of arrangements = 8! / (2! * 2!)
3. Simplify the expression to find the solution.
- 8! / (2! * 2!) = 40,320 / (2 * 2)
- = 40,320 / 4
- = 10,080.
Therefore, the number of distinguishable permutations of the word "EDUCATED" is 10,080.