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14.Probability
medium
Suppose $n \ge 3$ persons are sitting in a row. Two of them are selected at random. The probability that they are not together is
A
$1 - \frac{2}{n}$
B
$\frac{2}{{n - 1}}$
C
$1 - \frac{1}{n}$
D
None of these
Solution
(a) Let there be $n$ persons and $(n – 2)$ persons not selected are arranged in places stated above by stars and the selected $2$ persons can be arranged at places stated by dots (dots are $n – 1$ in number)
So the favourable ways are $^{n – 1}{C_2}$ and the total ways are $^n{C_2}$, so
$ \times \bullet \times \bullet \times \bullet \times \bullet \times \bullet \times $
$P = \frac{{^{n – 1}{C_2}}}{{^n{C_2}}} $
$= \frac{{(n – 1)\,!\,2\,!\,(n – 2)\,!}}{{(n – 3)\,!\,2\,!\,n\,!}} = \frac{{n – 2}}{n} = 1 – \frac{2}{n}$.
Standard 11
Mathematics
