Question

Five digit number is formed using the digits 0, 1, 2, 3, 4 and 5 without repetitions. Number of five digit numbers which are divisible by 10 is ________.

• A. 360
• B. 240
• C. 120
• D. 480
• E. 520

Hint:

Fix the last digit as 0 first.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
We need to form 5-digit numbers using the set of 6 digits \{0, 1, 2, 3, 4, 5\} without repetition. There are two conditions: 1. The number must be a 5-digit number, which means the first digit cannot be 0. 2. The number must be divisible by 10, which means the last digit must be 0.
Step 2: Key Formula or Approach
We will use the box method (or multiplication principle) to count the number of possibilities. We should always handle the most restrictive conditions first. In this case, the condition on the last digit is the most specific.
Step 3: Detailed Explanation
Let's set up 5 boxes to represent the five digits of the number:
\(\underline{\text{T-Th}} \ \underline{\text{Th}} \ \underline{\text{H}} \ \underline{\text{T}} \ \underline{\text{U}}\)
The available digits are \{0, 1, 2, 3, 4, 5\}.
1. Fill the most restricted position: The Units (U) place.
For the number to be divisible by 10, the units digit must be 0.
- Number of choices for the Units place = 1 (only the digit 0).
The digit 0 is now used. The remaining available digits are \{1, 2, 3, 4, 5\}.
2. Fill the next restricted position: The Ten Thousands (T-Th) place.
A 5-digit number cannot start with 0. - Since we have already used the digit 0 for the units place, this condition is automatically satisfied. Any of the remaining digits can be used.
- The remaining digits are \{1, 2, 3, 4, 5\}. There are 5 digits available.
- Number of choices for the Ten Thousands place = 5.
3. Fill the remaining positions.
- After filling the first and last places, we have \(5 - 1 = 4\) digits remaining.
- Number of choices for the Thousands (Th) place = 4.
- Now we have 3 digits remaining.
- Number of choices for the Hundreds (H) place = 3.
- Now we have 2 digits remaining.
- Number of choices for the Tens (T) place = 2.
4. Calculate the total number of possibilities.
Using the multiplication principle, the total number of 5-digit numbers is:
\[ \text{Total} = (\text{Choices for T-Th}) \times (\text{Choices for Th}) \times (\text{Choices for H}) \times (\text{Choices for T}) \times (\text{Choices for U}) \] \[ \text{Total} = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] This is also equivalent to fixing the last digit (1 way) and arranging the remaining 5 digits in the first 4 places, which is not correct. The correct way is: \[ \text{Total} = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] The calculation is \(P(5,4)\) for the first four digits and 1 for the last digit. Number of ways = \(P(5,4) \times 1 = \frac{5!}{(5-4)!} = 5! = 120\).
Step 4: Final Answer
The number of five-digit numbers divisible by 10 is 120.