Question

How many 6-digit numbers are there in all?

Answer 1

Hint:
Here, we have to use the concept of permutation for calculating the total number of 6-digit numbers in all. First, we will find the number of choices available for the first place digit of the 6-digit number. Then we will find the number of choices available for the rest of the digits of the 6-digit number by using the permutation. Then by solving this we will get the total number of 6-digit numbers in all.

Complete step by step solution:
It is given that the numbers are the 6 digit numbers.
So, we know that for the first place digits in the 6 digit number there are only 9 choices are available i.e. \[1, 2, 3, 4, 5, 6, 7, 8, 9\] as 0 cannot be in the first place digit because then it will become a 5 digit number.
Therefore, total number of permutation for the first digit in the 6 digit number \[ = {}^9{{\rm{P}}_1} = \dfrac{{9!}}{{{\rm{(9 - 1)}}!}} = 9\]
Now, we know that for the rest of the 5 digits in the 6 digit number there are 10 choices available i.e. \[0,1,2,3,4,5,6,7,8,9\]
Total number of permutation for all the remaining 5 digits in the 6 digit number \[ = {}^{10}{{\rm{P}}_1} = \dfrac{{10!}}{{{\rm{(10 - 1)}}!}} = 10\]
So, total number of 6 digit number in all \[ = 9 \times 10 \times 10 \times 10 \times 10 \times 10 = 900000\]

Hence, there are total \[9,00,000\] 6-digit numbers.

Note:
Permutations may be defined as the different ways in which a collection of items can be arranged. For example: The different ways in which the numbers 1, 2 and 3 can be grouped together, taken all at a time, are \[123,{\rm{ }}132,{\rm{ }}213{\rm{ }}231,{\rm{ }}312,{\rm{ }}321\].
Also we can calculate this using the basic formula
Total 6 digit number \[ = \] Largest 6 digit number \[ - \] Smallest 6 digit number \[ + 1\]
We know that the largest 6 digit number is 999999 and the smallest 6 digit number is 100000.
Therefore, substituting the value, we get
Total 6 digit number \[ = 999999 - 100000 + 1\]
Subtracting the terms, we get
Total 6 digit number \[ = 899999 + 1 = 900000\]
Hence, there are total \[9,00,000\] 6-digit numbers.