To solve the problem of finding the number of four-letter words consisting of two distinct letters and two alike letters taken from the word "SYLLABUS," we can follow these steps:
### Step 1: Identify the Letters
The word "SYLLABUS" consists of the following letters:
- S (occurs 2 times)
- Y (occurs 1 time)
- L (occurs 2 times)
- A (occurs 1 time)
- B (occurs 1 time)
- U (occurs 1 time)
From this, we can see that we have:
- 2 alike letters (either S or L)
- 2 distinct letters (from the remaining letters)
### Step 2: Choose the Distinct Letters
We need to choose 2 distinct letters from the available letters. The distinct letters available are S, Y, L, A, B, and U. However, since we need to consider that S and L can also be the alike letters, we will treat them as distinct for this step.
The total distinct letters we can choose from are:
- S, Y, L, A, B, U (total of 6 letters)
We will choose 2 distinct letters from these 6 letters. The number of ways to choose 2 distinct letters from 6 is given by the combination formula:
\[
\text{Number of ways to choose 2 distinct letters} = \binom{6}{2} = 15
\]
### Step 3: Choose the Alike Letter
Now, we need to choose one letter to be the alike letter. The possible alike letters can be either S or L. Therefore, we have:
- If we choose S as the alike letter, we can choose one of the remaining distinct letters (which we have already chosen).
- If we choose L as the alike letter, we can choose one of the remaining distinct letters (which we have already chosen).
Thus, we have 2 choices for the alike letter.
### Step 4: Calculate the Total Combinations
Now we multiply the number of ways to choose the distinct letters by the number of ways to choose the alike letter:
\[
\text{Total combinations} = \binom{6}{2} \times 2 = 15 \times 2 = 30
\]
### Step 5: Arrange the Letters
Now, we need to arrange the 4 letters (2 alike and 2 distinct). The arrangement of the letters can be done using the formula for permutations of multiset:
\[
\text{Number of arrangements} = \frac{4!}{2!} = \frac{24}{2} = 12
\]
### Step 6: Final Calculation
Finally, we multiply the total combinations by the number of arrangements:
\[
\text{Total number of 4-letter words} = 30 \times 12 = 360
\]
### Conclusion
Thus, the total number of four-letter words consisting of two distinct letters and two alike letters taken from the word "SYLLABUS" is **360**.
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