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What is the total number of different 13 -letter arrangements that can be formed using the letters in the word MATHEMATICIAN? Answer:

What is the total number of different 1313 -letter arrangements that can be formed using the letters in the word MATHEMATICIAN?\newlineAnswer:

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Full solution

Q. What is the total number of different 1313 -letter arrangements that can be formed using the letters in the word MATHEMATICIAN?\newlineAnswer:
  1. Frequency of Each Letter: Determine the frequency of each letter in the word MATHEMATICIAN.\newlineMM appears 22 times.\newlineAA appears 33 times.\newlineTT appears 22 times.\newlineHH appears 11 time.\newlineEE appears 11 time.\newline2200 appears 22 times.\newline2222 appears 11 time.\newline2244 appears 11 time.
  2. Calculate Total Factorial: Calculate the factorial of the total number of letters, which is 1313. \newline13!=13×12×11×10×9×8×7×6×5×4×3×2×113! = 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
  3. Factorial of Frequency: Calculate the factorial of the frequency of each letter to account for indistinguishable arrangements.\newlineM: 2!=2×12! = 2 \times 1\newlineA: 3!=3×2×13! = 3 \times 2 \times 1\newlineT: 2!=2×12! = 2 \times 1\newlineH: 1!=11! = 1\newlineE: 1!=11! = 1\newlineI: 2!=2×12! = 2 \times 1\newlineC: 1!=11! = 1\newlineN: 1!=11! = 1
  4. Permutations Formula: Use the formula for permutations of a multiset to find the total number of different arrangements.\newlineThe formula is:\newlineTotal arrangements = 13!(2!×3!×2!×1!×1!×2!×1!×1!)\frac{13!}{(2! \times 3! \times 2! \times 1! \times 1! \times 2! \times 1! \times 1!)}
  5. Perform Calculations: Perform the calculations using the values from the previous steps.\newlineTotal arrangements = 13!2!×3!×2!×1!×1!×2!×1!×1!\frac{13!}{2! \times 3! \times 2! \times 1! \times 1! \times 2! \times 1! \times 1!}\newlineTotal arrangements = (13×12×11×10×9×8×7×6×5×4×3×2×1)((2×1)×(3×2×1)×(2×1)×1×1×(2×1)×1×1)\frac{(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}{((2 \times 1) \times (3 \times 2 \times 1) \times (2 \times 1) \times 1 \times 1 \times (2 \times 1) \times 1 \times 1)}\newlineTotal arrangements = (13×12×11×10×9×8×7×6×5×4×3×2×1)(2×6×2×1×1×2×1×1)\frac{(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}{(2 \times 6 \times 2 \times 1 \times 1 \times 2 \times 1 \times 1)}\newlineTotal arrangements = (13×12×11×10×9×8×7×6×5×4×3×2×1)(48)\frac{(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)}{(48)}\newlineTotal arrangements = (13×12×11×10×9×8×7×6×5×4×3×1)(2)\frac{(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 1)}{(2)}\newlineTotal arrangements = (13×12×11×10×9×8×7×6×5×4×3)(2)\frac{(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3)}{(2)}\newlineTotal arrangements = 13×12×11×10×9×8×7×6×5×4×32\frac{13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3}{2}\newlineTotal arrangements = 10378368001037836800

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