The number of natural numbers lying between 1012 and 23421 that can...

Given number should be $4$ digit numbers between $1012 \text{\text{ and } }$ 23421 and can be formed using the number $2, 3, 4, 5, \text{\text{ and } }$ 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)$will be \left(6, 4, 3\right), \left(3, 4, 6\right), \left(2, 3, 6\right), \left(6, 3, 2\right), \left(3, 2, 4\right) \text{\text{ and } } \left(4, 2, 3\right)$

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 $= 6$