The total number of coins is:
\[
100 \, (\text{50 paise}) + 50 \, (\text{₹1}) + 20 \, (\text{₹2}) + 10 \, (\text{₹5}) = 180
\]
The number of ₹5 coins is 10. Therefore, the probability of picking a ₹5 coin is:
\[
P(\text{₹5 coin}) = \frac{10}{180} = \frac{1}{18}
\]
Thus, the correct answer is option (4).