Question

How many $5 - $digit numbers can be formed using$\left( {0 - 9} \right)$?

Answer 1

Hint:
Here we need to just multiply all the number of ways by which the then thousandth place, thousandth place, hundredth, tenth and one place can be filled. Here the trick to be kept in mind is that the first digit cannot be zero as it would not form the five digit number instead would form a four digit number.

Complete step by step solution:
Here we need to find the number of $5 - $ digit numbers that can be formed using $\left( {0 - 9} \right)$
This can be understood with one example in a better way. If we need to form the three digit number using the digits $0, 1, 2$ then we know that we need to fill the one’s place, ten’s place and the hundredth place with the help of these three digits. So let us see the number of ways by which we can do so.
In the hundredth place only $1,2$ can be occupied as we need to form a three digit number. If we add $0$ there also, then it would become a two digit number and the rest two places can occupy all three digits. Hence we can say that the total number of ways will be $\left( {2 \times 3 \times 3} \right) = 18{\text{ ways}}$
Now similarly here we need to form a $5 - $digit numbers that can be formed using$\left( {0 - 9} \right)$
So we can say that we need to fill the five places to get the five digit number. Here we need to just multiply all the number of ways by which the then thousandth place, thousandth place, hundredth, tenth and ones place can be filled.
We know that the number cannot start with $0$ hence there are only $9$ digits which are $\left( {1 - 9} \right)$ that can occupy the starting of the number. So there are $9$ ways to do it.
In the rest four places we can add any digits from $\left( {0 - 9} \right)$ so there are $10$ ways to do so.

Hence we can say that total ways are:
Ways $ = $$9 \times 10 \times 10 \times 10 \times 10$ which gives us $90000$

Note:
Here the student must know that whenever we are not told about whether the digits can be repeated or not, we need to count all the digits which may include all the repeating as well as non –repeating. Moreover we need to see that if $0$ can be one of the parts of the number that is to be formed, we need to remember that it must not occupy the first place of the number to be formed.