Four numbers are chosen at random (without replacement) from the set \1, 2, 3, , 20\ .Statement-1: The probability that the chosen numbers when arrang - Mathematics MCQ AIEEE 2026 | ExamTest.live | ExamTest.live

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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 form an AP, then the set of all possible values of common difference is {\pm 1, \pm 2, \pm 3, \pm 4, \pm 5} .

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

Solution & Step-by-step Explanation

Total ways to choose 4 numbers = (4 20) = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845 .Possible A.P.s: d = 1 : {1, 2, 3, 4}, \dots, {17, 18, 19, 20} (17 cases) d = 2 : {1, 3, 5, 7}, \dots, {14, 16, 18, 20} (14 cases) d = 3 : {1, 4, 7, 10}, \dots, {11, 14, 17, 20} (11 cases) d = 4 : {1, 5, 9, 13}, \dots, {8, 12, 16, 20} (8 cases) d = 5 : {1, 6, 11, 16}, \dots, {5, 10, 15, 20} (5 cases) d = 6 : {1, 7, 13, 19}, {2, 8, 14, 20} (2 cases)Total A.P. cases = 17 + 14 + 11 + 8 + 5 + 2 = 57 .Prob = \frac{57}{4845} = \frac{1}{85} . (Statement 1 is True).Possible common differences are {1, 2, 3, 4, 5, 6} . (Statement 2 is False as it misses 6).

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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 form 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 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.

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