Question

The number of seven-digit numbers that can be formed by using the digits \(1,2,3,5,7\) such that each digit is used at least once, is:

• A. \(15400\)
• B. \(17800\)
• C. \(16800\)
• D. \(29400\)

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We need to form 7-digit numbers using 5 distinct digits where every digit is used at least once.
Since we have 7 slots and 5 digits, two digits must be extra. This leads to two distinct cases for the distribution of frequencies of the digits.
Step 2: Key Formula or Approach:
Identify possible frequency partitions of 7 into 5 parts where each part \(\geq 1\):
Case 1: One digit is used 3 times, others used 1 time each. (Partition: 3, 1, 1, 1, 1)
Case 2: Two digits are used 2 times each, others used 1 time each. (Partition: 2, 2, 1, 1, 1)
Step 3: Detailed Explanation:
Case 1: One digit appears thrice, others once.
1. Select which digit is repeated 3 times: \({}^{5}C_{1} = 5\) ways.
2. Arrange the 7 digits (with one digit repeated thrice): \(\frac{7!}{3! \times 1! \times 1! \times 1! \times 1!} = \frac{5040}{6} = 840\) ways.
Total for Case 1 = \(5 \times 840 = 4200\).
Case 2: Two digits appear twice, others once.
1. Select which two digits are repeated 2 times each: \({}^{5}C_{2} = 10\) ways.
2. Arrange the 7 digits (with two digits repeated twice): \(\frac{7!}{2! \times 2! \times 1! \times 1! \times 1!} = \frac{5040}{4} = 1260\) ways.
Total for Case 2 = \(10 \times 1260 = 12600\).
Total Number of 7-digit numbers = \(4200 + 12600 = 16800\).
Step 4: Final Answer:
The number of such seven-digit numbers is 16800.