The number of natural numbers lying between $1012$ and $23421$ that can be formed using the digits $2,3,4,5,6$ (repetition of digits is not allowed) and divisible by $55$ is _____.

Given number should be $4$ digit numbers between $1012 \& 23421$ and can be formed using the number $2, 3, 4, 5, \& 6$ and should be divisible by $55$

CASE I: When number has $4$ digit, 

Now for divisibility by $55$, number should be divisible by $5$ and $11$ both

Also, for divisibility by $11$ for below figure,

Now fixing $d=5$ the total possible cases for $\left(a,b,c\right)$ will be $\left(6,4,3\right),\left(3,4,6\right),\left(2,3,6\right),\left(6,3,2\right),\left(3,2,4\right) \& \left(4,2,3\right)$

CASE II:

When number has $5$ digits.

Let the number be $abcde$.

Here $e$ is fixed at $5$. So the other $4$ digits must be from $2,3,4,6$

No such number is possible because even the least number formed is greater than $23421$.

Hence, the total number of required numbers $=6$