Question

The number of integers greater than $6,000$ that can be formed using the digits $3, 5, 6, 7$ and $ 8$ without repetition is

• A. 216
• B. 192
• C. 120
• D. 72

Answer 1

The Correct Option is B

To find the number of integers greater than 6,000 that can be formed using the digits 3, 5, 6, 7, and 8 without repetition, we need to consider the possible arrangements of these digits. Here’s a detailed solution:

1. First, identify that a number greater than 6,000 formed with the given digits must have 5 digits or more.
2. Let's consider only 5-digit numbers since using all these digits gives the maximum possible formations:
3. The first digit, which determines whether the number is greater than 6,000, must be either 6, 7, or 8.
   • If the first digit is 6, the remaining four digits (3, 5, 7, 8) can be arranged in 4! ways.
   • If the first digit is 7, the remaining four digits (3, 5, 6, 8) can be arranged in 4! ways.
   • If the first digit is 8, the remaining four digits (3, 5, 6, 7) can be arranged in 4! ways.
4. Calculate the total number of 5-digit numbers: $$ 3 \times 4! = 3 \times 24 = 72 $$
5. Since we are also interested in numbers with fewer digits (but these cannot be formed because the minimum 4-digit number with these digits is 6,735), re-evaluate any other possibility.

Hence, the total number of integers that can be formed, greater than 6,000 and utilizing the digits 3, 5, 6, 7, 8 exactly once, is 72.