Step 1: Identify the vowels and consonants ✨

The word MATHEMATICS has 11 letters 🅰️🔤

• Vowels (🟢): A, E, A, I
• Consonants (🔵): M, T, H, M, T, C, S

Step 2: Treat the vowels as one unit 🧩

Since the vowels (A, E, A, I) must stay together, we treat them as one unit.
Now, instead of 11 letters, we have 8 units:

• (AEAI) as one unit 🟢
• M, T, H, M, T, C, S (the 7 consonants) 🔵

Step 3: Arrange the 8 units 🔄

The 8 units can be arranged in:

8!2!2!\frac{8!}{2!2!} 2!2!8!​

(Since M appears twice and T appears twice, we divide by 2! to avoid duplicates.)

403204=10080\frac{40320}{4} = 10080440320​=10080
Step 4: Arrange the vowels 🎶

The vowels A, E, A, I inside their unit can be arranged in:

4!2!=242=12\frac{4!}{2!} = \frac{24}{2} = 122!4!​=224​=12

(Since A appears twice, we divide by 2!)

Step 5: Find the total number of ways ✨
10080×12=12096010080 \times 12 = 12096010080×12=120960
🎯 Final Answer:

The letters of MATHEMATICS can be arranged in 120,960 ways while keeping the vowels together! 🎊