The total number of numbers greater than $4, 00, 000$ that can be formed by using the digits $0, 2, 2, 4, 4, 5$ is

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Solution

Suppose, the six-digit number is denoted by $a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}$
For the number to be greater than $4, 00, 000,$
$a_{1}$ should be either $4$ or $5.$

If $a_{1}$ is $4,$ then $a_{2} a_{3} a_{4} a_{5} a_{6}$ can be arranged in $\frac{5!}{2!} = 60$ ways
If $a_{1}$ is $5,$ then $a_{2} a_{3} a_{4} a_{5} a_{6}$ can be arranged in $\frac{5!}{2! 2!} = 30$ ways

$∴$ Total number of required numbers $= 90$

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