Question

How many numbers formed using the digits 3, 2, 0, 4, 3, 2, 3 exceed one million?

Answer 1

A number that exceeds one million is to be formed from the digits 3, 2, 0, 4, 3, 2, 3.
Then the numbers should be any number of 7 digits that can be formed from these digits.
Also among the given numbers 2 repeats twice and 3 repeats thrice.
∴ Required number of numbers
= Total number of arrangements possible among these digits − number of arrangements of 7 digits which begin with 0.
= `(7!)/(2!3!)-(6!)/(2!3!)`

= `(7xx6xx5xx4xx3!)/(2xx3!)-(6xx5xx4xx3!)/(2xx3!)`
= 7 × 6 × 5 × 2 −6 × 5 × 2
= 6 × 5 × 2(7 − 1)
= 60 × 6
= 360
∴ 360 numbers that exceed one million can be formed with the digits 3, 2, 0, 4, 3, 2, 3.