There is a five-volume dictionary among $$50$$ books arranged on a shelf in 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 :
A.
$$\frac{1}{5}$$
B.
$$\frac{1}{{{5^{50}}}}$$
C.
$$\frac{1}{{{{50}^5}}}$$
D.
none of these
Answer :
none of these
Solution :
The number of ways of arranging $$50$$ books$$ = {}^{50}{P_{50}} = 50!.$$
The number of ways of choosing places for the five volume dictionary is $${}^{50}{C_5}$$ and the number of ways of arranging the remaining $$45$$ books $$ = {}^{45}{P_{45}} = \left( {45} \right)!.$$
Thus the number of favourable ways is $$\left( {{}^{50}{C_5}} \right)\left( {45!} \right).$$
Hence the probability of the required event
$$\eqalign{
& = \frac{{\left( {{}^{50}{C_5}} \right)\left( {45!} \right)}}{{50!}} \cr
& = \left( {\frac{{50!}}{{5!\,45!}}} \right)\left( {\frac{{45!}}{{50!}}} \right) \cr
& = \frac{1}{{5!}} \cr
& = \frac{1}{{120}} \cr} $$
Releted MCQ Question on Statistics and Probability >> Probability
Releted Question 1
Two fair dice are tossed. Let $$x$$ be the event that the first die shows an even number and $$y$$ be the event that the second die shows an odd number. The two events $$x$$ and $$y$$ are:
Two events $$A$$ and $$B$$ have probabilities 0.25 and 0.50 respectively. The probability that both $$A$$ and $$B$$ occur simultaneously is 0.14. Then the probability that neither $$A$$ nor $$B$$ occurs is
The probability that an event $$A$$ happens in one trial of an experiment is 0.4. Three independent trials of the experiment are performed. The probability that the event $$A$$ happens at least once is
If $$A$$ and $$B$$ are two events such that $$P(A) > 0,$$ and $$P\left( B \right) \ne 1,$$ then $$P\left( {\frac{{\overline A }}{{\overline B }}} \right)$$ is equal to
(Here $$\overline A$$ and $$\overline B$$ are complements of $$A$$ and $$B$$ respectively).
A.
$$1 - P\left( {\frac{A}{B}} \right)$$
B.
$$1 - P\left( {\frac{{\overline A }}{B}} \right)$$
C.
$$\frac{{1 - P\left( {A \cup B} \right)}}{{P\left( {\overline B } \right)}}$$
D.
$$\frac{{P\left( {\overline A } \right)}}{{P\left( {\overline B } \right)}}$$