यदि $0,1,3,5$ और $7$ अंकों द्वारा $5000$ से बड़ी चार अंकों की संख्या का यादृच्छ्धा निर्माण किया गया हो तो पाँच से भाज्य संख्या के निर्माण की क्या प्रायिकता है जब, अंकों की पुनरावृत्ति नहीं की जाए ?

When the digits are repeated

since four - digit numbers greater than $5000$ are formed, the leftmost digit is either $7$ or $5 .$

The remaining $3$ places can be filled by any of the digits $0,\,1,\,3,\,5,$ or $7$ as repetition of digits is allowed.

$\therefore$ Total number of $4\, -$ digit numbers greater than $5000=2 \times 5 \times 5 \times 5-1$

$=250-1=249$

$[$ In this case, $5000$ can not be counted; so $1 $ is subtracted $]$

A number is divisible by $5$ if the digit at its units place is either $0$ or $5$.

$\therefore$ Total number of $4 \,-$ digit numbers greater than $5000$ that are divisible by $5=$ $2 \times 5 \times 5 \times 2-1=100-1=99$

Thus, the probability of forming a number divisible by $5$ when the digits are repeated is $=$ $\frac{99}{249}=\frac{33}{83}$