JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Probability

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Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference, is:

(A) $\frac{15}{101}$
(B) $\frac{5}{101}$
(C) $\frac{5}{33}$ ✓ Correct
(D) $\frac{10}{99}$

Answer: (C)

Solution

Three numbers form an arithmetic progression exactly when the middle one is the average of the outer two, which needs the two chosen extreme terms to share parity. Among

11

consecutive natural numbers there are

6

of one parity and

5

of the other, so the number of same-parity pairs that fix an AP is

(2 6) + (2 5) = 15 + 10 = 25

. The total number of ways to pick three numbers is

(3 11) = 165

, so the probability is

\frac{25}{165} = \frac{5}{33}

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Single correct option

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