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What is the total number of different 11-letter arrangements that can be formed using the letters in the word MEASUREMENT? Answer:

What is the total number of different 1111-letter arrangements that can be formed using the letters in the word MEASUREMENT?\newlineAnswer:

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Q. What is the total number of different 1111-letter arrangements that can be formed using the letters in the word MEASUREMENT?\newlineAnswer:
  1. Determine Letter Frequencies: Determine the frequency of each letter in the word MEASUREMENT.\newlineThe word MEASUREMENT has 1111 letters with the following frequency of each letter:\newlineM=2M = 2, E=3E = 3, A=1A = 1, S=1S = 1, U=1U = 1, R=1R = 1, N=1N = 1, T=1T = 1.
  2. Use Permutations Formula: Use the formula for permutations of a multiset to calculate the number of different arrangements.\newlineThe formula is:\newlineNumber of arrangements = n!n1!×n2!×…×nk!\frac{n!}{n_1! \times n_2! \times \ldots \times n_k!}\newlinewhere nn is the total number of letters, and n1n_1, n2n_2, …\ldots, nkn_k are the frequencies of each distinct letter.\newlineFor MEASUREMENT, this becomes:\newlineNumber of arrangements = 11!2!×3!×1!×1!×1!×1!×1!×1!\frac{11!}{2! \times 3! \times 1! \times 1! \times 1! \times 1! \times 1! \times 1!}
  3. Calculate Factorials: Calculate the factorial of each number.\newline11!=11×10×9×8×7×6×5×4×3×2×111! = 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\newline2!=2×12! = 2 \times 1\newline3!=3×2×13! = 3 \times 2 \times 1\newlineSince the factorials of 11 are all 11, we can ignore them in the calculation.
  4. Simplify Factorial Expressions: Simplify the factorial expressions.\newline11!=3991680011! = 39916800\newline2!=22! = 2\newline3!=63! = 6
  5. Calculate Number of Arrangements: Substitute the factorial values into the formula and calculate the number of arrangements.\newlineNumber of arrangements = 39916800(2×6)\frac{39916800}{(2 \times 6)}\newlineNumber of arrangements = 3991680012\frac{39916800}{12}\newlineNumber of arrangements = 33264003326400

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