Question

A dice is rolled three times, The probability of getting a larger number than the previous number is $\frac{k}{54}$.Find the value of $k$

Answer 1

Total possible outcomes $=n\left(S\right)=6\times 6\times 6=216$.
Favourable outcomes : $(1,2,3),(1,2,4),(1,2,5),(1,2,6),(1,3,4),(,1,3,5),(1,3,6),(1,4,5),(1,4,6),(1,5,6),$

$(2,3,4),(2,3,5),(2,3,6),(2,4,5),(2,4,6),(2,5,6),(3,4,5),(3,4,6),(3,5,6),(4,5,6)$
Thus,favorable outcomes $=n\left(E\right)=20$
Hence, probability $=\frac{20}{216}=\frac{5}{54}\Rightarrow k=5$