Question

question
what is the total number of different 12-letter arrangements that can be formed using the letters in the word neighborhood?
answer attempt 2 out of 2
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Explanation:

Step1: Count the letters and their frequencies

The word NEIGHBORHOOD has 12 letters. The letter frequencies are: N - 1, E - 1, I - 1, G - 1, H - 2, B - 1, O - 3, R - 1, D - 1.

Step2: Use the permutation formula for multi - sets

The formula for permutations of a multi - set is \(\frac{n!}{n_1!n_2!\cdots n_k!}\), where \(n\) is the total number of objects, and \(n_i\) are the frequencies of each distinct object. Here \(n = 12\), \(n_1=2\) (for H) and \(n_2 = 3\) (for O). So the number of arrangements is \(\frac{12!}{2!×3!}\).

Step3: Calculate the factorial values

\(12! = 479001600\), \(2! = 2\), \(3! = 6\). Then \(\frac{479001600}{2×6}=39916800\).

Answer:

\(39916800\)