Five numbers x₁, x₂, x₃, x₄, x₅ are randomly selected from the numbers 1, 2, 3, …, 18 and are arranged in the increasing order such that x₁ < x₂ < x₃ < x₄…

Answer: (c) 1/(68) Total choices binom185 = 8568. With x₂ = 7, x₄ = 11: x₁ from 1 to 6 (6 ways), x₃ from 8 to 10 (3 ways), x₅ from 12 to 18 (7 ways), so 6 × 3 × 7 = 126 ways. Probability = (126)/(8568) = 1/(68). Correct: (c)

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Mathematics20261 markMCQMatrices

Choose the correct option.

Five numbers x₁, x₂, x₃, x₄, x₅ are randomly selected from the numbers 1, 2, 3, …, 18 and are…

Five numbers $x_1, x_2, x_3, x_4, x_5$ are randomly selected from the numbers $1, 2, 3, \dots, 18$ and are arranged in the increasing order such that $x_1 < x_2 < x_3 < x_4 < x_5$. What is the probability that $x_2 = 7$ and $x_4 = 11$? (Application)

(a)$\frac{26}{51}$
(b)$\frac{3}{104}$
(c)$\frac{1}{68}$
(d)$\frac{1}{34}$

Answer

Official answer key

Correct option: (c)

Answer: (c) $\frac{1}{68}$ Total choices $\binom{18}{5} = 8568$. With $x_2 = 7$, $x_4 = 11$: $x_1$ from $1$ to $6$ (6 ways), $x_3$ from $8$ to $10$ (3 ways), $x_5$ from $12$ to $18$ (7 ways), so $6 \times 3 \times 7 = 126$ ways. Probability $= \frac{126}{8568} = \frac{1}{68}$.