<s>[INST] Given digits $2,2,3,3,4,4,4,4$ how many distinct $4$ digit numbers greater than $3000$ can be formed? Given digits $2,2,3,3,4,4,4,4$ how many distinct $4$ digit numbers greater than $3000$ can be formed?
one of the digits which can be formed is $4444$
$4$ digit numbers greater than $3000$, which consists of only $2's$ and $4's$ are $4224$, $4242$, $4244$, $4422$, $4424$, $4442$
is there a well defined technique to solve this question.  
 [/INST] You want to make a $4$ digit number. Now lets do it with Permutation-Combination.We have 4 places to fill different numbers.
First place can have either $3$ or $4$. So we have two choices. Lets analyse it.
If first place is $3$ - so we have to choose 3 digits from $(2,2,3, 3,4,4,4,4)$. So any place can have either 2 or 3 or 4, but $(222,333)$ is not possible, because we have just two $3$'s  and two $2$'s
So total = $3 \times 3\times 3 - 2 = 25$
If first place is $4$ - so now we have to choose 3 digits from $(2,2,3, 3,3,4,4,4)$. So any place can have either $2$ or $3$ or $4$, but $(222)$ is not possible, because we have just two $2$'s
So total = $3 \times 3\times 3 - 1 = 26$
So, total choices = $25+26 = 51$, which is your answer.
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