Four numbers are chosen at random ( without replacement ) from th… - Vidyadip

MCQ

Four numbers are chosen at random ( without replacement ) from the set $\{1,2,3,..,20\}$

Statement $-1 :$ The probability that the chosen numbers when arranged in some order will form an $A.P.$ is $\frac{1}{{85}}$ .

Statement $-2 :$ If the four chosen numbers form an $A.P.$, then the set of all possible values of common difference is $\left( { \pm 1, \pm 2, \pm 3, \pm 4, \pm 5} \right)$ છે.

A
Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is a correct explanation for Statement $-1$
B
Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is not a correct explanation for Statement $-1$
C
Statement $-1$ is false, Statement $-2$ is true
✓
Statement $-1$ is true, Statement $-2$ is false

✓

Answer

Correct option: D.

Statement $-1$ is true, Statement $-2$ is false

d
(c) : Number of $A.P's$ with common difference $1=17$

Number of $A.P.'s$ with common difference $2=14$

Number of $A.P.'s$ with common difference $3=11$

Number of $A.P.'s$ with common difference $4=8$

Number of $A.P.'s$ with common difference $5=5$

Number of $A.P.'s$ with common difference $6 = \frac{2}{{57}}$

The total number of ways $n(S) = {\,^{20}}{{\rm{C}}_4}$

The desired probability $= \frac{{57}}{{^{20}{{\rm{C}}_4}}}$

$ = \frac{{57 \times 24}}{{20 \times 19 \times 18 \times 17}} = \frac{1}{{85}}$

Now statement- $2$ is false and statement- $1$ is true.

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