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:
For a set of $n$ distinct non-zero digits, each digit appears in each place (units, tens, hundreds, etc.) $(n-1)!$ times across all possible permutations.
Key Formula or Approach:
Sum of all $n$-digit numbers formed using $n$ distinct digits is:
\[ (\text{Sum of digits}) \times (n-1)! \times (11\dots1 \text{ repeated } n \text{ times}) \]

Step 2: Detailed Explanation:
1. Identify the values:
Digits are $\{3, 5, 7\}$, so $n = 3$.
Sum of digits = $3 + 5 + 7 = 15$.
2. Calculate frequency of each digit in each position:
Frequency = $(3-1)! = 2! = 2$.
3. Calculate the total sum:
\[ \text{Sum} = 15 \times 2 \times (111) \]
\[ \text{Sum} = 30 \times 111 = 3330 \]

Step 3: Final Answer:
The sum of all numbers in $S$ is 3330.