Question

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

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

Answer 1

Answer: (B)

Statement-1 is true, Statement-2 is false

Answer 2

$N(S) = \,^{20}C_4$ 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 $Prob. = \frac{17 +14+ 11+ 8+ 5+ 2}{^{20}C_{4}} = \frac{1}{85}.$