Step 1: Understanding the Concept:
When specific items must be "together", we treat them as a single block or entity. We then arrange this block along with the remaining letters, accounting for repetitions using the multinomial permutation formula.
Step 2: Key Formula or Approach:
Total arrangements = (Arrangements of entities including the block) \(\times\) (Internal arrangements of the block).
Formula: \( \frac{n!}{p!q!...} \) for repeated items.
Step 3: Detailed Explanation:
In the word "MATHEMATICS":
Consonants: M, T, H, M, T, C, S (7 letters: M repeats twice, T repeats twice).
Vowels: A, E, A, I (4 letters: A repeats twice).
1. Treat the 4 vowels [AAEI] as one single block.
2. Total entities to arrange = 7 consonants + 1 vowel block = 8 entities.
Arrangements of 8 entities = \( \frac{8!}{2! \times 2!} \) (dividing by \(2!\) for M and \(2!\) for T).
\[ \frac{40320}{4} = 10080 \]
3. Internal arrangements of the vowel block [AAEI] = \( \frac{4!}{2!} \) (dividing by \(2!\) for A).
\[ \frac{24}{2} = 12 \]
4. Total permutations = \( 10080 \times 12 = 120960 \).
Step 4: Final Answer:
The total number of ways is 120960.