How many 6 - digit numbers can be formed by using the digits 4, 5, 0, 3, 4, 5?

To find: number of 6 digit

0 cannot be in the first place because that would make a 5 - digit number

Total number of 6 - digit numbers = Total number of number possible - Number of numbers with 0 at the first place

Total number of numbers possible $=\frac{6 !}{2 ! 2 !}=180$

Number of numbers with 0 at first place $=\frac{5 !}{2 ! 2 !}=30$

$\Rightarrow$ Number of 6 - digit numbers $=180-30=150$

150 six - digit numbers are possible