Two numbers 'a' and 'b' are selected successively without replacement

Question:

Two numbers 'a' and 'b' are selected successively without replacement in that order from the integers 1 to 10 . The probability that $\frac{a}{b}$ is an integer, is

(a) $\frac{17}{45}$

(b) $\frac{1}{5}$

(c) $\frac{17}{90}$

(d) $\frac{8}{45}$

Solution:

We have a set of natural numbers from 1 to 10 where andare two variables which can take values from 1 to 10.

So, total number of possible combination of $a$ and $b$ so that $\left(\frac{a}{b}\right)$ is a fraction without replacement are:

$\left(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \frac{1}{10}\right)$

Similarly we have 9 such sets of 10 elements each. So total number of possible combination,

$=(9)(10)$

$=90$

Now the possible combination which makes $\left(\frac{a}{b}\right)$ an integer without replacement are-

$=\left(\frac{2}{1}, \frac{3}{1}, \frac{4}{1}, \frac{5}{1}, \frac{6}{1}, \frac{7}{1}, \frac{8}{1}, \frac{9}{1}, \frac{10}{1}, \frac{4}{2}, \frac{6}{2}, \frac{6}{3}, \frac{8}{2}, \frac{8}{4}, \frac{9}{3}, \frac{10}{2}, \frac{10}{5}\right)$

$=17$

Therefore the probability that $\left(\frac{a}{b}\right)$ is an integer,

$=\frac{\text { Possible combination which makes }\left(\frac{a}{b}\right) \text { an integer }}{\text { Total possible combination of }\left(\frac{a}{b}\right)}$

$=\frac{17}{90}$

The correct answer is option (c)

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