Final answer:
The number of different permutations for the word MATHEMATICIAN, considering the repetition of letters, is calculated using the formula 13! / (2! × 3! × 2! × 2!), which results in 362,880 different words.
Step-by-step explanation:
The question involves calculating the number of different words (permutations) that can be made by re-arranging the letters of the word MATHEMATICIAN. To find this, we start by counting the total number of letters in the word, which is 13. However, we must consider the repetition of certain letters: 'M' appears twice, 'A' appears three times, 'T' appears twice, and 'I' appears two times. The formula for permutations when there are repeating items is n! / (p1! × p2! × ... × pk!), where n is the total number of items, and pi is the number of repetitions of the ith item.
To solve this, we calculate 13! (13 factorial) and then divide that by the factorial of the counts of each repeating letter: 2! for 'M', 3! for 'A', 2! for 'T', and 2! for 'I'. Therefore, the number of different words that can be formed is 13! / (2! × 3! × 2! × 2!) which simplifies to 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × (3 × 2 × 1) / (2 × 1 × 6 × 2 × 1 × 2 × 1). Performing the calculations gives us the answer, which is option (c) 362,880.