The number of words (with or without meaning) that can be formed from all the letters of the word "L

Q: Wait, why do we use the Gap Method instead of just subtracting the cases where vowels are together from the total?

That is a brilliant question! While the 'Total minus Together' method works perfectly when you have only two items to keep apart, it becomes a nightmare when you have more complex constraints. The Gap Method is your 'universal key' for 'never together' problems. It is cleaner, less prone to double-counting, and builds a stronger intuition for how objects occupy space.

Q: Why did we divide by $2!$ for the consonants? I thought we just arranged them.

Remember, in combinatorics, if you swap two identical items (like the two $T$'s in 'LETTER'), the word remains exactly the same. If we treated them as distinct, we would be overcounting. Dividing by $2!$ is our way of 'dividing out' those redundant, invisible swaps.

Q: Does the order of the vowels matter when we place them in the gaps?

In this specific case, the vowels are both $E$. Since they are identical, swapping them doesn't create a new word. However, if the vowels were different (like $A$ and $E$), you would multiply by $2!$ to account for their internal arrangements. Always check if your items are distinct or identical!

Q: Why are there 5 gaps for 4 consonants? Shouldn't it be 4?

Think of the consonants as four people standing in a line: $\_ C_1 \_ C_2 \_ C_3 \_ C_4 \_$. You have a space before the first, spaces between each, and a space after the last. That gives you $4 + 1 = 5$ possible slots. If you place a vowel in the end slots, they are still separated from other vowels by at least one consonant!

Q: Is it possible to get a different answer if I place the vowels first?

If you place the vowels first, you would have to account for the fact that the vowels themselves might be separated by consonants, which makes the logic much more complex. The Gap Method is designed to make the consonants the 'anchors' so that the vowels have no choice but to be separated. Stick to the Gap Method—it is the most robust path to the truth.

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

Sigma Percentile · Accepted Answer

### The Art of Separation: Mastering the Gap Method

Welcome, future engineer. Today, we are not just solving a permutation problem; we are learning how to impose order on chaos.

The word 'LETTER' might seem simple, but it hides a classic combinatorial challenge: the 'never together' constraint. When we are told that vowels must never touch, we are essentially being asked to build a structure where the consonants act as buffers.

Let us embark on this journey of logical construction.

### Phase 1: Deconstructing the Word

First, look at the raw material. The word is $LETTER$. We have $6$ letters in total.

But wait—look closer. We have $2$ vowels ($E, E$) and $4$ consonants ($L, T, T, R$). The presence of identical letters is a trap for the unwary, but for us, it is just a detail to manage with precision.

We have identified our players: the vowels are $E$ and $E$, and the consonants are $L, T, T, R$.

### Phase 2: The Consonant Foundation

To ensure the vowels never touch, we must first place the consonants. Think of these consonants as the pillars of a bridge. We arrange them first, and then we look for the 'safe zones'—the gaps—where we can tuck the vowels away.

We have $4$ consonants: $L, T, T, R$. If we were to arrange these as if they were distinct, we would have $4!$ ways. But we have two identical $T$'s.

To correct for this, we divide by $2!$. The number of ways to arrange our consonant pillars is:

$$	ext{Ways} = \frac{4!}{2!} = \frac{24}{2} = 12$$

We have successfully built our foundation. We have $12$ distinct ways to arrange these four consonants.

### Phase 3: The Geometry of Gaps

Now, imagine these $4$ consonants standing in a row. Where can we place our vowels so they never touch? We can place them in the spaces before the first consonant, between any two consonants, or after the last one.

Let us visualize this:

$$\_ C_1 \_ C_2 \_ C_3 \_ C_4 \_$$

Counting these underscores, we see $5$ available gaps. This is the beauty of the Gap Method. By placing our $2$ vowels into these $5$ gaps, we guarantee that there will always be at least one consonant between them.

We need to choose $2$ gaps out of $5$, which is a classic combination problem:

$$\binom{5}{2} = \frac{5 	imes 4}{2 	imes 1} = 10$$

### Phase 4: The Final Assembly

We are almost there. We have arranged the consonants in $12$ ways, and we have chosen the gaps for our vowels in $10$ ways. Finally, we must consider the vowels themselves.

Since both vowels are $E$, there is only $\frac{2!}{2!} = 1$ way to arrange them within the chosen gaps. To find the total number of words, we multiply these independent choices together:

$$	ext{Total} = 12 	imes 10 	imes 1 = 120$$

### The Takeaway

Look at that number: **$120$**. It is not just an answer; it is the result of a systematic, logical process.

Whenever you face a 'never together' constraint in your JEE exams, do not panic. Build your pillars, identify your gaps, and let the mathematics do the heavy lifting.

You have mastered the Gap Method today—a tool that will serve you well in every permutation challenge you face. Keep practicing, keep questioning, and most importantly, keep falling in love with the elegance of the logic.

Animated Solution for Mathematics - Permutations and Combinations: 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

Enter Numerical Value:

Visualized Solution

Analyzing the word $L E T T E R$

Identifying Vowels and Consonants

The Gap Method Strategy

Arranging the Consonants

Handling Identical Consonants

Calculating Consonant Permutations

Visualizing the Gaps

Choosing Gaps for Vowels

Calculating Gap Selections

Arranging the Vowels

Final Total Calculation

The Final Answer

The Sigma Insight: Linear Permutations

The Art of Separation

Phase 1

Phase 2

Phase 3

Phase 4

The Takeaway

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