Question

Number of natural numbers that can be formed using digits 1, 2, 3, 4, 5, 6, 7 each exactly once so that digits 3, 4 and 5 are always in the middle is equal to:

• A. 24
• B. 144
• C. 5040
• D. 720

Hint:

When fixing certain digits in a given position, calculate the arrangements of the remaining digits separately and multiply the results accordingly.

Answer 1

The Correct Option is B

Step 1: Fixing digits 3, 4, and 5 in the middle.
Since the digits 3, 4, and 5 are always to be in the middle, they are fixed in three positions in the middle of the number. 
Step 2: Choosing other digits.
The remaining positions will be filled by the remaining digits: 1, 2, 6, and 7. These digits can be arranged in \( 4! \) (4 factorial) ways. \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \] Step 3: Conclusion.
Thus, the total number of natural numbers that can be formed is \( 24 \times 6 = 144 \) since there are 6 ways to arrange the digits 3, 4, and 5 in the middle positions.