The number of words, with or without meaning, that can be formed by taking 4 letters at a time from

Q: Why can't we just pick 4 letters and then arrange?

Because the selection process depends on the frequency of letters. We must first ensure we have the correct number of identical and distinct letters, which requires a structured selection process rather than a random pick.

Q: Why do we divide by 2! in the arrangement step?

Because swapping identical letters doesn't create a new word. We must divide by the factorial of the number of identical items to avoid overcounting.

Q: What if we picked the same letter for the distinct part?

The problem specifies 'distinct' letters, so we must exclude the letter used for the pair to ensure the two distinct letters are truly different from the pair and from each other.

Question

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike, is

Sigma Percentile · Accepted Answer

### Analyzing the Building Blocks

To solve this problem, we first identify the composition of the word **SYLLABUS**. The letters are: S, Y, L, L, A, B, U, S.

We observe the following frequencies:
*   **S**: 2
*   **L**: 2
*   **Y, A, B, U**: 1 each

There are 6 distinct types of letters available: {S, L, Y, A, B, U}.

### Selecting the Letters

We need to form a 4-letter word consisting of one pair of identical letters and two distinct letters.

First, we choose the pair. Since there are two types of letters that appear twice (S and L), the number of ways to choose the pair is:
$$^2C_1 = 2$$

Next, we must choose two distinct letters from the remaining types. Having used one type for the pair, we have $6 - 1 = 5$ types remaining. The number of ways to choose 2 distinct letters from these 5 is:
$$^5C_2 = \frac{5 	imes 4}{2 	imes 1} = 10$$

### Arranging the Letters

Now, we calculate the number of ways to arrange these 4 chosen letters. If all 4 letters were distinct, there would be $4! = 24$ arrangements.

Because we have one pair of identical letters, we must divide by $2!$ to account for the overcounting of identical arrangements:
$$	ext{Arrangements} = \frac{4!}{2!} = \frac{24}{2} = 12$$

### Final Calculation

To find the total number of 4-letter words, we multiply the number of selection ways by the number of arrangement ways:
$$	ext{Total} = 2 	imes 10 	imes 12$$

**Total = 240**

Animated Solution for Mathematics - Permutations and Combinations: The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike, is

Enter Numerical Value:

Visualized Solution

Analyze the word $S Y LL A B U S$

Frequency of Letters

Define the Selection Strategy

Step 1: Selecting the Alike Pair

Computing Alike Pair Selection

Step 2: Selecting the Distinct Letters

Computing Distinct Letters Selection

Step 3: Arrangement Setup

Computing the Arrangements

Final Calculation Setup

The Final Answer

The Sigma Insight: Combinations and Selection

Analyzing the Building Blocks

Selecting the Letters

Arranging the Letters

Final Calculation

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