Introduction / Context:
This question is about counting the number of fixed length digit sequences formed from a given set of digits when repetition is allowed. We are asked how many 5 digit telephone numbers can be formed using digits from 0 to 9 with no extra restrictions.
Given Data / Assumptions:
- Digits available: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
- We need to form 5 digit numbers.
- Repetition of digits is allowed.
- A telephone number is considered valid even if it starts with 0.
Concept / Approach:
There are 10 possible digits for each position in the 5 digit number and each position choice is independent if repetition is allowed. Therefore, the total number of different 5 digit sequences is given by a power of 10.
Step-by-Step Solution:
Step 1: Consider the first digit. It can be any of the 10 digits from 0 to 9, so there are 10 possibilities.
Step 2: The second digit can again be any of the 10 digits from 0 to 9.
Step 3: The same reasoning applies to the third, fourth and fifth digits.
Step 4: Using the multiplication rule of counting, total number of 5 digit numbers = 10 * 10 * 10 * 10 * 10 = 10^5.
Step 5: Compute 10^5 = 100000, often written as 1,00,000.
Verification / Alternative check:
Each 5 digit number corresponds to a unique sequence of choices from the set of 10 digits. Since there are 10 possibilities for each of the 5 positions, the total of 10^5 combinations is logical and straightforward.
Why Other Options Are Wrong:
Values like 59,049 or 6561 are powers of 3 or combinations that arise in other contexts (for example, 3^10 or 3^8), and 3439 is a random value which does not match the structure of 10^5 for this problem.
Common Pitfalls:
Sometimes learners incorrectly impose a condition that the first digit cannot be 0, which would give 9 * 10^4 instead of 10^5. In this specific question, that extra restriction is not present because it explicitly refers to telephone numbers, which can be treated as fixed length sequences, not standard decimal integers.
Final Answer:
The number of possible 5 digit telephone numbers under these conditions is
1,00,000.
Discussion & Comments