Question

How many words can be formed using all the letters of the word TUESDAY such that all the vowels are together?

• A. 720
• B. 120
• C. 360
• D. 1440

Hint:

"Vowels together" formula: $(n - v + 1)! \times v!$, where $n$ is total letters and $v$ is the number of vowels (provided no letters repeat).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
When certain items must be "together," we treat them as a single entity or "block." We then arrange this block along with the remaining letters, and finally, we arrange the letters {inside} the block.
Step 2: Key Formula or Approach:
1. Identify Vowels and Consonants.
2. Treat vowels as one unit.
3. Total arrangements = (Arrangement of units) $\times$ (Internal arrangement of vowels).
Step 3: Detailed Explanation:
1. Word: TUESDAY (Total 7 letters).
2. Vowels: U, E, A (3 letters).
3. Consonants: T, S, D, Y (4 letters).
4. Treat (U, E, A) as one unit. Now we have 4 consonants + 1 unit = 5 units to arrange.
5. Arrangement of 5 units = $5! = 120$.
6. Internal arrangement of 3 vowels = $3! = 6$.
7. Total arrangements = $120 \times 6 = 720$.
Step 4: Final Answer:
The total number of words formed is 720.