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

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 digits numbers = 420 - 60 = 360