Correct option is C
Given:
- Digits available: 1, 2, 3, 4, 5, 6
- We need to form 4-digit numbers
- The string 123 must appear together and in the same order
- No digit repeats
Concept Used:
Treat "123" as a single block since it must appear together.
Now, we are choosing 4-digit numbers, and one chunk is already "123" (3 digits together), so we only need to choose one more digit from the remaining digits.
Solution:
Step 1: Remaining digits after using 1, 2, 3 → {4, 5, 6}
We need to pick 1 digit from these 3.
So, number of ways to choose 1 more digit = 3
Step 2: Total digits to arrange = 123 + 1 new digit = 4 digits
We need to find how many ways to place "123" as a block along with the chosen digit.
- For each such 4-digit number:
- "123" can be placed in 2 ways:
- At the start: "123X"
- At the end: "X123"
→ So, for each selected extra digit, there are 2 positions
Step 3: Final calculation:
- Choose 1 digit from {4,5,6} = 3 ways
- For each, 2 placements = 3 × 2 = 6 ways
Correct Option: (C) 6
