Question

Four numbers are chosen at random (without replacement) from the set $\{1,2,3,...,20\}$.
Statement-l: The probability that the chosen numbers when arranged in some order will form an A.P. is $\frac{1}{85}$.
Statement-2: lf the four chosen numbers form an A.P., then the set of all possible values of common difference is $\{\pm 1,\pm 2,\pm 3,\pm 4,\pm 5\}$.

• A. Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation for Statement-1
• B. Statement-1 is true, Statement-2 is false
  
• C. Statement-1 is false, Statement-2 is true
• D. Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation for Statement-1

Answer 1

The correct option is B Statement-1 is true, Statement-2 is false

$N\left(S\right){=}^{20}{C}_4$
Statement-l: common difference is $1$; total number of cases $=17$ common difference is $2$; total number of cases $=14$ common difference is $3$; total number of cases $=11$ common difference is $4$; total number of cases $=8$; common difference is $5$; total number of cases $=5$ common difference is $6$; total number of cases $=2$
Probability $=\frac{17+14+11+8+5+2}{20C_4}=\frac{1}{85}$ .