Find the number of numbers, greater than a million,

Question:

Find the number of numbers, greater than a million, that can be formed with the digits 2, 3, 0, 3, 4, 2, 3.

Solution:

One million (1,000,000) consists of 7 digits.

We have digits 2, 3, 0, 3, 4, 2 and 3.

Numbers formed by arranging all these seven digits $=\frac{7 !}{2 ! 3 !}$

But, these numbers also include the numbers whose first digit is 0.

This is invalid as in that case the number would be less than a million.

Total numbers in which the first digit is fixed as $0=$ Permutations of the remaining 6 digits $=\frac{6 !}{2 ! 3 !}$

Numbers that are greater than 1 million $=\frac{7 !}{2 ! 3 !}-\frac{6 !}{2 ! 3 !}=360$