Question

Number of seven digits numbers which can be formed by using all the digits 1, 2, 3, 4, 5 with at least one digit repeated is _____.

• A. 16200
• B. 15600
• C. 16800
• D. 14800

Answer 1

The Correct Option is C

Step 1: Analyze the problem constraints.
We need to form 7-digit numbers using digits 1, 2, 3, 4, 5. The question implies that all these 5 digits must appear at least once in the number. Since we have 7 spots and only 5 available digits, repetition is mandatory.


Step 2: Identify cases where all 5 digits are used.
There are two possible distributions for the 7 digits to include all 5 specific numbers:
Case 1: One digit is used 3 times, and the other four digits are used 1 time each.
Case 3: Two digits are used 2 times each, and the other three digits are used 1 time each.


Step 3: Calculate permutations for Case 1.
Choose 1 digit to be repeated 3 times out of 5: $^5C_1$.
Arrange these 7 digits (where one repeats 3 times): $\frac{7!}{3!}$.
Total = $5 \times \frac{5040}{6} = 5 \times 840 = 4200$.


Step 4: Calculate permutations for Case 2.
Choose 2 digits to be repeated 2 times each out of 5: $^5C_2$.
Arrange these 7 digits (where two repeat 2 times each): $\frac{7!}{2!2!}$.
Total = $10 \times \frac{5040}{4} = 10 \times 1260 = 12600$.


Step 5: Sum the cases.
Total numbers = $4200 + 12600 = 16800$.
The answer is 16800 (Option 3).