Consider the digits in set `S=(1,2,3,4)`. All four-digit number formed using digits in `S` are arranged ascending order. Which of the following is/are correct?

Options

A. The number 3412 is at 177 th
B. the number 3423 is at 183th position
C. the number 3412 is at 178th position
D. the number at 110th position 2342

Explanation

To solve the problem, we need to determine the position of specific four-digit numbers formed using the digits from the set \( S = \{1, 2, 3, 4\} \) when arranged in ascending order. We will analyze the given numbers one by one.

### Step-by-Step Solution

1. **Identify the digits and total combinations**:
The digits available are \( 1, 2, 3, 4 \). Since we are forming four-digit numbers using all these digits, the total number of combinations can be calculated as:
\[ 4! = 24 \]
This means there are 24 unique four-digit numbers that can be formed using these digits.

2. **List all four-digit numbers**:
To find the position of a specific number, it helps to list all possible combinations in ascending order. The combinations are:
- 1234
- 1243
- 1324
- 1342
- 1423
- 1432
- 2134
- 2143
- 2314
- 2341
- 2413
- 2431
- 3124
- 3142
- 3214
- 3241
- 3412
- 3421
- 4123
- 4132
- 4213
- 4231
- 4312
- 4321

3. **Find the position of the number 3412**:
- We count how many numbers come before 3412.
- All numbers starting with 1: 6 combinations (1234, 1243, 1324, 1342, 1423, 1432).
- All numbers starting with 2: 6 combinations (2134, 2143, 2314, 2341, 2413, 2431).
- All numbers starting with 3 and second digit less than 4: 2 combinations (3124, 3142).
- Now we reach numbers starting with 34. The combinations are:
- 3412 (the number we are interested in).
- So, the position of 3412 is:
\[ 6 + 6 + 2 + 1 = 15 \]

4. **Find the position of the number 3423**:
- Count numbers before 3423:
- All numbers starting with 1: 6 combinations.
- All numbers starting with 2: 6 combinations.
- All numbers starting with 3 and second digit less than 4: 2 combinations (3124, 3142).
- All numbers starting with 34 and second digit less than 2: 0 combinations.
- Now we reach numbers starting with 342. The combinations are:
- 3421 (1 combination).
- 3423 (the number we are interested in).
- So, the position of 3423 is:
\[ 6 + 6 + 2 + 1 + 1 = 16 \]

5. **Find the position of the number 2342**:
- Count numbers before 2342:
- All numbers starting with 1: 6 combinations.
- All numbers starting with 2 and second digit less than 3: 2 combinations (2134, 2143).
- Now we reach numbers starting with 23. The combinations are:
- 2314 (1 combination).
- 2341 (1 combination).
- 2342 (the number we are interested in).
- So, the position of 2342 is:
\[ 6 + 2 + 1 + 1 + 1 = 11 \]

### Final Positions
- The position of **3412** is **15**.
- The position of **3423** is **16**.
- The position of **2342** is **11**.