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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
$\frac{1}{{5\,!}}$
$\frac{{50\,!}}{{55\,!}}$
$\frac{1}{{{{50}^5}}}$
None of these
Solution
(a) 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 $20\,\,!$ ways.
$\therefore $ Favourable ways $ = {}^{25}{C_5}.\,20\,\,! = \frac{{25\,\,!\,.\,20\,\,!}}{{5\,\,!\,\,20\,\,!}} = \frac{{25\,\,!}}{{5\,\,!}}$
Also total number of ways $ = 25\,\,!$
$\therefore $ Probability $ = \frac{{25\,\,!}}{{5\,\,!\,\,25\,\,!}} = \frac{1}{{5\,\,!}}$.
