Question

How many $7-digit$ numbers can be formed using the digits $1,2,0,2,4,2$ and $4$?

Answer 1

There are 7 digits 1, 2, 0, 2, 4, 2, 4 in which 2 occurs 3 times, 4 occurs 2 times.

Number of 7 digit numbers = $\frac{7!}{3!\times 2!}=420$

But out of these 420 numbers, there are some numbers which begin with '0' and they are not 7-digit numbers.

The number of such numbers beginning with '0' = $\frac{6!}{3!\times 2!}=60$

Hence, the required number of 7 digit numbers = 420 - 60 = 360.