In how many ways can the word MATHEMATICS be arranged?

Analysis

To find the number of ways to arrange the letters of a word, we use the formula for permutations of a multiset (arrangements with repeated items). The formula is:

where:

• $n$ is the total number of letters.
• $n_1, n_2, \dots$ are the frequencies of the repeated letters.

Concept

When some items in a set are identical, swapping them does not create a new unique arrangement. Therefore, we divide the total factorial ($n!$) by the factorial of the count of each repeated letter to remove these redundant arrangements.

Step-by-Step Solution

1. Count the total number of letters ($n$): M-A-T-H-E-M-A-T-I-C-S There are 11 letters in total. So, $n = 11$.

2. Identify the frequency of repeated letters: • M appears 2 times. • A appears 2 times. • T appears 2 times. • H appears 1 time. • E appears 1 time. • I appears 1 time. • C appears 1 time. • S appears 1 time.

3. Apply the formula: Substitute the values into the permutation formula: $$\text{Arrangements} = \frac{11!}{2! \times 2! \times 2!}$$(We divide by$2!$for M,$2!$for A, and$2!$ for T).

4. Simplify: The expression is $\frac{11!}{2!2!2!}$.

Conclusion

Comparing our result to the given options, we can see that it matches option C.

Exam Tip: Always write out the word and tally the frequency of each letter carefully before plugging them into the formula. A common mistake is forgetting to count the letters that appear only once (though they don't affect the denominator since $1! = 1$, it helps ensure you haven't missed any letters).