There is a five-volume dictionary among 50 books arranged on a shelf in a random order. If the volumes are not necessarily kept side-by side, the probability that they occur in increasing order from left to right is

none of these

The total number of ways of arranging 50 books in shelf is ${^{50}P}_{50}=50!$ Out of 50 places, 5 places for the five-volume dictionary can be chosen in ${^{50}C}_5$ ways. In the remaining 45 places the remaining 45 books can be arranged in ${^{45}P}_{45}=45!$ ways. In the five places five volumes of dictionary can be arranged in increasing order in one way only.

So, favourable number of ways $= {^{50}C}_5 × 45!$

Hence, required probability $=\frac{^{50}C_5 × 45!}{50!}=\frac{1}{120}$.