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 $$={}^9{\mathrm{P}}_1={\displaystyle \frac{9!}{(9-1)!}}=9$$
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 $$={}^{10}{\mathrm{P}}_1={\displaystyle \frac{10!}{(10-1)!}}=10$$
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 $$123,132,213231,312,321$$.
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.