A die is cast twice and a coin is tossed thrice. Find the probability that the die will turn a 5 each time and the coin will turn
a head every time.

Correct option is D

Given:

A die is cast twice and a coin is tossed thrice.
​Formula Used:
Probability of an event = $$\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$​​

​Solution:​
Probability of rolling a 5 on a die = $$\frac{1}{6}$$​ (since there are 6 faces on the die).
Probability of getting a head on a coin = $$\frac{1}{2}$$​ (since there are 2 faces on the coin).
Probability of getting a 5 on the die for both rolls:
P (5 on die twice) = $$\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$$​​
Probability of getting heads on the coin for all three tosses:
P (heads on coin thrice) =  $$\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$$​​
Total probability (since the events are independent):
P (desired outcome) = $$\frac{1}{36} \times \frac{1}{8} = \frac{1}{288}$$​​

Thus, the correct option is (d) $$\frac{1}{288}$$​
​
Alternate Solution: 

Possible Outcome =  (Total Faces)^{number of cast}​

Total number of possible outcomes when coin is cast thrice = $$2^3$$ = 8

​Total number of possible outcomes when dice is cast twice = 6² = 36

Total probability = $$\frac{1}{8 \times 36}$$ =$$\frac{1}{288}$$​​