Question

There are 5 volumes of Mathematics among 25 books. They are arranged on a shelf in random order. The probability that the volumes of Mathematics stand in increasing order from left to right (the volumes are not necessarily kept side by side) is

• A. $\frac { 1 } { 5 ! }$
• B. $\frac { 50 ! } { 55 ! }$
• C. $\frac { 1 } { 50 ^ { 5 } }$
• D. None of these

Answer 1

Answer: (A)

$\frac { 1 } { 5 ! }$

Answer 2

Five places for 5 volumes of Mathematics from 25 places of books can be select in ${ } ^ { 25 } C _ { 5 }$ ways.

In these places, we are not to permute the 5 volumes since order of these 5 volumes is fixed i.e.,

$v _ { 1 } , v _ { 2 } , v _ { 3 } , v _ { 4 } , v _ { 5 }$

Remaining 20 books can be arranged in ways.

$\therefore$ Favourable ways $= { } ^ { 25 } C _ { 5 } \cdot 20 ! = \frac { 25 ! .20 ! } { 5 ! 20 ! } = \frac { 25 ! } { 5 ! }$

Also total number of ways $= 25 !$

$\therefore$ Probability $= \frac { 25 ! } { 5 ! 25 ! } = \frac { 1 } { 5 ! }$ .