Question

Let $S$ be the set of all 3-digit numbers containing the digits 3, 5 and 7 without repetition. The sum of all numbers in $S$ is

• A. 3330
• B. 2220
• C. 4590
• D. 1110
• E. 4440

Hint:

For 3 digits $x, y, z$, the sum is always $222(x+y+z)$. Here, $222 \times 15 = 3330$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We are forming 3-digit numbers using 3 distinct digits without repetition.
There are \(3! = 6\) such numbers. We need to find their sum.
Step 2: Key Formula or Approach:
The sum of all \(n\)-digit numbers formed by \(n\) distinct digits without repetition is given by:
\[ \text{Sum} = (n-1)! \times (\text{Sum of digits}) \times (111\dots \text{n times}) \] Step 3: Detailed Explanation:
The given digits are 3, 5, and 7.
Number of digits \(n = 3\).
Sum of the digits = \(3 + 5 + 7 = 15\).
Apply the formula:
\[ \text{Sum} = (3-1)! \times (15) \times 111 \] \[ \text{Sum} = 2! \times 15 \times 111 = 2 \times 15 \times 111 \] \[ \text{Sum} = 30 \times 111 = 3330 \] Step 4: Final Answer:
The sum is 3330.