JEE Main · 2010 · 11 Apr 2010 · Offline · Probability

Four numbers are chosen at random (without replacement) from the set

{1, 2, 3, \dots, 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
(B)Statement-1 is true, Statement-2 is false✓ Correct
(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: (B)

Solution

Choosing four numbers from twenty can be done in

(4 20) = 4845

ways. Four chosen numbers form an arithmetic progression for some common difference

d \geq 1

, and the count of such progressions is

20 - 3 d

, namely

17, 14, 11, 8, 5, 2

for

d = 1, 2, 3, 4, 5, 6

, a total of

57

. The probability is

\frac{57}{4845} = \frac{1}{85}

, so Statement-1 is true. Statement-2 lists the possible common differences as only

\pm 1

through

\pm 5

, but

d = 6

also works, for example

2, 8, 14, 20

, so it is incomplete and false.

JEE Main · 2010 · 11 Apr 2010 · Offline · Single correct option