Find the number of arrangements of the letters of the word independence

General
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find the number of arrangements of the letters of the word independence

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Find the Number of Arrangements of the Letters of the Word “Independence”

The word “independence” consists of 12 letters. However, these letters are not all distinct. To find the number of unique arrangements, we must account for the repeated letters. Here’s how it breaks down:

  • Frequency of each letter:
    • I: 1
    • N: 3
    • D: 2
    • E: 4
    • P: 1
    • C: 1

To find the total number of unique arrangements, we use the formula for permutations of a multiset:

\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}

Where:

  • n! is the factorial of the total number of letters.
  • n_1!, n_2!, \ldots, n_k! are the factorials of the frequencies of the repeating letters.

Step-by-Step Calculation

  1. Calculate the Total Factorial:

    The total number of letters, n, is 12, so we start by calculating 12!.

    12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479,001,600

  2. Calculate Factorials of Frequency:

    • For ‘I’ with frequency 1: 1! = 1
    • For ‘N’ with frequency 3: 3! = 6
    • For ‘D’ with frequency 2: 2! = 2
    • For ‘E’ with frequency 4: 4! = 24
    • For ‘P’ with frequency 1: 1! = 1
    • For ‘C’ with frequency 1: 1! = 1

  3. Substitute into the Multiset Formula:

    We then substitute all these factorial values into the formula:

    \frac{12!}{1! \times 3! \times 2! \times 4! \times 1! \times 1!} = \frac{479,001,600}{1 \times 6 \times 2 \times 24 \times 1 \times 1}

  4. Calculate the Denominator:

    First, calculate the product of the factorials in the denominator:

    1 \times 6 \times 2 \times 24 \times 1 \times 1 = 288

  5. Divide the Total Number by the Denominator:

    Finally, divide the total factorial by the denominator to find the number of unique arrangements:

    \frac{479,001,600}{288} = 1,663,200

Thus, the number of distinct arrangements of the letters in the word “independence” is 1,663,200.

Key Points and Summary

  • The problem involves arranging letters in a word with multiple repeated characters.
  • Use the formula for permutations of a multiset to account for repetition.
  • Calculate the total factorial for all letters, then divide by the factorials of the frequencies of each repeating letter.
  • Resulting calculation gave us the solution: 1,663,200 unique arrangements.

Remember that this follows the principle of dividing the permutation of a set by the permutations of its indistinguishable subsets, allowing us to find the number of unique orderings. If you have further questions about permutations or any related concepts, feel free to ask! @anonymous7