The total number of words (with or without meaning) that can be formed out of the letters of the wor

Q: Wait, why can't I just use the formula $12P4$?

That is the most common trap! The formula $12P4$ assumes that all 12 letters are distinct. However, in 'DISTRIBUTION', if you swap the first 'I' with the second 'I', you haven't created a new word. Because of these identical letters, we must use a case-by-case approach to account for the reduced number of unique arrangements.

Q: How do I know if I have missed a case?

Think of it as partitioning the integer 4 (the length of the word). The possible partitions are: 4 (all different), 3+1 (three alike, one different), 2+2 (two alike, two alike), and 2+1+1 (two alike, two different). Since we have exhausted all ways to sum to 4, we have covered all possible scenarios.

Q: Why do we divide by factorials in the arrangement step?

When you have identical items, the number of distinct arrangements is reduced. For example, if you have 4 letters where 2 are identical, the total permutations $4!$ count the arrangements of the identical items as distinct. Dividing by $2!$ 'cancels out' those redundant arrangements, giving you the true count of unique words.

Question

The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to _______.

Sigma Percentile · Accepted Answer

### The Art of Counting: Unraveling DISTRIBUTION

Imagine you are a cryptographer tasked with decoding a message. You are given the word 'DISTRIBUTION' and asked to form new words of length $4$.

It sounds simple, but the moment you look at the letters, you realize this is not just a simple counting exercise—it is a puzzle of symmetry and repetition.

### Phase 1: The Inventory

Before we touch a single formula, we must understand our raw materials. We have the word 'DISTRIBUTION'. Let us count our inventory:

- 'I' appears $3$ times.
- 'T' appears $2$ times.
- 'D', 'S', 'R', 'B', 'U', 'O', 'N' each appear $1$ time.

In total, we have $12$ letters, but only $9$ distinct types of letters. This is the crucial realization. If we treat all letters as distinct, we will fall into the trap of overcounting; we must be systematic.

### Phase 2: The Four Pillars of Selection

Since we are picking $4$ letters, we must categorize our selection based on the patterns of repetition. We have four mutually exclusive cases:

**Case 1: 3 Alike, 1 Different**

We need a triplet. Only 'I' has $3$ copies, so we select the triplet in $\binom{1}{1}$ way. We need one more letter from the remaining $8$ distinct types, which is $\binom{8}{1}$ ways.

The arrangement is given by:
$$ \frac{4!}{3!} = 4 $$

Thus, the total for this case is $1 	imes 8 	imes 4 = 32$ words.

**Case 2: 2 Alike, 2 Alike**

We need two pairs. Only 'I' and 'T' have at least $2$ copies, so we must select both pairs in $\binom{2}{2} = 1$ way.

The arrangement of $4$ letters with two pairs is:
$$ \frac{4!}{2!2!} = 6 $$

Thus, the total for this case is $1 	imes 6 = 6$ words.

**Case 3: 2 Alike, 2 Different**

We need one pair and two distinct letters. We choose the pair from $\{I, T\}$ in $\binom{2}{1} = 2$ ways, and then we choose $2$ letters from the remaining $8$ distinct types in $\binom{8}{2} = 28$ ways.

The arrangement is:
$$ \frac{4!}{2!} = 12 $$

Thus, the total for this case is $2 	imes 28 	imes 12 = 672$ words.

**Case 4: All 4 Different**

We choose $4$ distinct letters from the $9$ available types in $\binom{9}{4} = 126$ ways. Since they are all distinct, the arrangement is $4! = 24$.

Thus, the total for this case is $126 	imes 24 = 3024$ words.

### Phase 3: The Synthesis

Now, we bring it all together. The beauty of this method is that because our cases are mutually exclusive, we can simply sum them up to find the total number of words:

$$ Total = 32 + 6 + 672 + 3024 = 3734 $$

And there you have it. By breaking the chaos of the word 'DISTRIBUTION' into these four logical pillars, we have arrived at the answer: **$3734$**.

It is a testament to the power of systematic thinking. Never let a complex problem intimidate you; just break it down, case by case, and the solution will reveal itself.

Animated Solution for Mathematics - Permutations and Combinations: The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to _______.

Enter Numerical Value:

Visualized Solution

Analysis of 'DISTRIBUTION'

Letter Frequencies

Strategy: Case-by-Case Analysis

Case 1: $3$ Alike, $1$ Different

Case 2: $2$ Alike, $2$ Alike

Case 3: $2$ Alike, $2$ Different

Case 4: All $4$ Different

Final Summation

The Sigma Insight: Combinations and Selection

The Art of Counting

Phase 1

Phase 2

Phase 3

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