# Three Integers are  randomly chosen without replacement from to first 20 postive integer. Find the problality that this 3 intgless from an A.P Option 1) $\frac{5}{38}$ Option 2) $\frac{1}{19}$ Option 3) $\frac{3}{38}$ Option 4) $\frac{2}{19}$

Probability of occurrence of an event -

Let S be the sample space then the probability of occurrence of an event E is denoted by P(E) and it is defined as

$P\left ( E \right )=\frac{n\left ( E \right )}{n\left ( S \right )}$

$P\left ( E \right )\leq 1$

$P(E)=\lim_{n\rightarrow\infty}\left(\frac{r}{n} \right )$

- wherein

Where n repeated experiment and E occurs r times.

Total Sum A.P series is

$2+4+---18$

$=\frac{9}{2}\left [ 2+18 \right ]=\frac{9}{2}\times 20$

$9\times 10=90$

and  3 number are chosen from

20 number is  $20_{C_{3}}$

$\therefore$ Required Probability =$\therefore \frac{90}{20_{C_{3}}}=\frac{90}{1410}$

$=\frac{9}{141}=\frac{3}{38}$

Option 1)

$\frac{5}{38}$

Option is incorrect

Option 2)

$\frac{1}{19}$

Option is incorrect

Option 3)

$\frac{3}{38}$

Option is correct

Option 4)

$\frac{2}{19}$

Option is incorrect

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