Question

The total number of 8 digit numbers is
[a] 9,000
[b] 9,00,000
[c] 9,00,00,000
[d] None of these

Answer 1

Hint: Use the fundamental principle of counting, which states that if a task A can be done in m ways and another task B can be done in n ways, the number of ways of doing both the tasks m and n is mn. Use the fact that the formation of an 8 digit number is equivalent to filling 8 decimal places with digits 0-9 such that the leftmost place is filled with a non-zero digit. Hence find the total number of 8 digit numbers. Alternatively, use the fact that the smallest 8 digit number is 1,00,00,000 and the largest 8 digit number is 9,99,99,999. Use the fact that the number of numbers between a and b both included = b-a+1. Hence find the total number of 8 digit numbers.

Complete step-by-step answer:
Fundamental principle of counting: According to this principle if a task A can be done in m ways and another task B can be done in N ways, then the number of ways in which both the tasks can be done is nm.
Now consider the case of determining how many 8 digit numbers are there. This is equivalent to finding the number of ways in which an 8 digit number can be created.
Now for the creation of an 8 digit number, we need to fill the decimal places with digits 0-9 in such a way that the leftmost decimal place is filled with a non-zero digit.
Hence all the places except the leftmost place can be filled in 10 ways each. The leftmost place can be filled in 9 ways(digits 1-9).
Hence by the fundamental principle of counting, the total number of ways in which an 8 digit number can be created is equal to $9\times 10\times 10\cdots \left( \text{ Seven 10s} \right)=9\times {{10}^{7}}=9,00,00,000$
Hence option [c] is correct.

Note: Alternative Solution:
The smallest 8 digit number is 1,00,00,000 and the largest 8 digit number is 9,99,99,999
All other 8 digit numbers are in between these two numbers.
We know that the number of numbers between a and b both included = b-a+1
Hence the total number of 8 digit number $=9,99,99,999-1,00,00,000+1=8,99,99,999+1=9,00,00,000$, which is the same as obtained above
Hence option [c] is correct.