This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the number of ways the word MATHEMATICIAN can be arranged, we use the formula for permutations with repetitions. Step 1: Count the total number of letters in the word. The word is MATHEMATICIAN. Total number of letters, n = 13. Step 2: Identify the repeated letters and their frequencies. • M appears 2 times (n_M = 2) • A appears 3 times (n_A = 3) • T appears 2 times (n_T = 2) • H appears 1 time • E appears 1 time • I appears 2 times (n_I = 2) • C appears 1 time • N appears 1 time Step 3: Apply the permutation formula for words with repeated letters. The formula is: (n!)/(n_1! n_2! ·s n_k!) where n is the total number of letters, and n_1!, n_2!, , n_k! are the factorials of the frequencies of the repeated letters. Step 4: Substitute the values and calculate. Number of arrangements = (13!)/(2! · 3! · 2! · 2!) Calculate the factorials: 13! = 6,227,020,800 2! = 2 × 1 = 2 3! = 3 × 2 × 1 = 6 Substitute these values into the formula: Number of arrangements = (6,227,020,800)/(2 × 6 × 2 × 2) Number of arrangements = (6,227,020,800)/(48) Number of arrangements = 129,729,600 Comparing this result with the given options: A) 6794800 ways B) 2664910 ways C) 6227020800 ways D) 129729600 ways The calculated number of arrangements matches option D. The final answer is D) 129729600 ways.