Question

The number of words (with or without meaning) that can be formed from all the letters of the word "LETTER" in which vowels never come together is _ _ _ _ _ .

Hint:

For "vowels never together", always: 1) Arrange consonants 2) Place vowels in gaps This avoids double counting.

Answer 1

Correct Answer: 120

Step 1: Understanding the Concept:
To ensure certain items never come together, we use the "Gap Method." We first arrange the consonants and then place the vowels in the empty spaces (gaps) created between and around the consonants.
Step 2: Key Formula or Approach:
Permutations with repetition: \(\frac{n!}{p!q!\dots}\)
Step 3: Detailed Explanation:
1. Word: "LETTER".
Letters: L, E, T, T, E, R.
Consonants: L, T, T, R (4 letters, with two T's identical).
Vowels: E, E (2 letters, both identical).
2. Arrange the consonants:
\[ \text{Ways} = \frac{4!}{2!} = \frac{24}{2} = 12 \text{ ways} \]
3. Create gaps between consonants: \(\wedge C \wedge C \wedge C \wedge C \wedge\)
There are \(4 + 1 = 5\) gaps.
4. Place the vowels (E, E) in these 5 gaps. Since the vowels are identical, the order of placement within chosen gaps does not matter.
\[ \text{Ways} = \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \text{ ways} \]
5. Total words \(= \text{Arrangement of consonants} \times \text{Placement of vowels}\):
\[ \text{Total} = 12 \times 10 = 120 \]
Step 4: Final Answer:
The number of required words is 120.