Question

How many 6 digit numbers are there in all?

Answer 1

Hint: We solve this problem by arranging the digits we have in 6 places to form the 6 digit numbers.
We have the digits from 0 to 9 that count to 10 digits.
We assume that there are 6 boxes that represent the 6 digits of a 6 digit number. Then we take the possibilities for each box and then calculate the total number of numbers by using the permutations.

Complete answer:
We are asked to find the number of 6 digit numbers that are there in total.
Let us assume that there are 6 boxes that represent the 6 places of a 6 digit number as follows

Now, let us take the number of possibilities for the first box.
We know that there are 10 digits from 0 to 9
But we know that the first place should not be 0 because if the first place occupies 0 then it becomes 5 digit number
So, we get the number of possibilities for first box as 9 by excluding the 0
Now, by taking the number of possibilities for first box then we get

Here, we can see that we need all 6 digit numbers which means that the digits can be repeated
So, we can say that all the remaining boxes have possibilities of 10 digits as the digits can be repeated.
By taking the number of possibilities in other boxes then we get

Now, let us assume that the total number of 6 digit numbers as \[N\]
We know that the total number of numbers is the permutations of each box’s possibility
By using the above condition we get the total number of 6 digit numbers as
\[\begin{align}
  & \Rightarrow N=9\times 10\times 10\times 10\times 10\times 10 \\
 & \Rightarrow N=9,00,000 \\
\end{align}\]
Therefore we can conclude that there are total of 900000 numbers of 6 digit numbers.


Note:
We can solve this problem in another method.
We are asked to find the number of 6 digit numbers.
Let us take the least 6 digit number as
\[\Rightarrow r=1,00,000\]
Now, let us take the largest 6 digit number as
\[\Rightarrow n=9,99,999\]
Let us assume that the total number of 6 digit numbers as \[N\]
We know that the number of numbers from \[n\] to \[r\] is given as \[n-r+1\]
By using the above result we get the number of 6 digit numbers as
\[\begin{align}
  & \Rightarrow N=n-r+1 \\
 & \Rightarrow N=9,99,999-1,00,000+1 \\
 & \Rightarrow N=9,00,000 \\
\end{align}\]
Therefore we can conclude that there are a total of 900000 numbers of 6 digit numbers.