A box contains $15$ marbles out of which $4$ are white, $5$ are red and $6$ are blue. Three balls are to be drawn at random from the bag. What is the probability that all of them are red?

The total number of marbles in the box is:
$$n(S)=4+5+6=15$$

The number of ways to draw $3$ marbles out of $15$ is given by ${}^{15}{\text{C}}_3$:

$${}^{15}{\text{C}}_3=\frac{15\times 14\times 13}{3\times 2\times 1}=5\times 7\times 13=455$$

We want all $3$ drawn marbles to be red. Since there are $5$ red marbles, the number of favorable ways is given by ${}^5{\text{C}}_3$:

$${}^5{\text{C}}_3=\frac{5\times 4\times 3}{3\times 2\times 1}=10$$

Therefore, the required probability $P(E)$ is:

$$P(E)=\frac{n(E)}{n(S)}=\frac{10}{455}=\frac{2}{91}$$