Q42 / 90 · JEE Main 2010 · Mathematics

Four numbers are chosen at random (without replacement) from the set $\{1,2,3, \ldots ., 20\}$. Statement-1: The probability that the chosen numbers when arranged in some order will form an AP is $\frac{1}{85}$. Statement-2: If the four chosen numbers from an AP, 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 not the correct explanation for Statement-1
  • Correct answer
    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

Solution

$$ \mathrm{N}(\mathrm{S})={ }^{20} \mathrm{C}_4 $$ Statement-1: 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 $=2$ $$ \text { Prob. }=\frac{17+14+11+8+5+2}{{ }^{20} \mathrm{C}_4}=\frac{1}{85} \text {. } $$

Hence, the correct answer is option: 'B'.

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