An integer is chosen at random from the integers 1, $2,3, \ldots, 50$. The probability that the chosen integer is a multiple of atleast one of 4,6 and 7 is

JEE Main · 2024 · 29 Jan 2024 · Shift 2 · Probability

An integer is chosen at random from the integers 1,

2, 3, \dots, 50

. The probability that the chosen integer is a multiple of atleast one of 4,6 and 7 is

Answer: (B)

Let

A, B, C

be the events that a number from

1

50

is a multiple of

4, 6, 7

respectively.

∣ A ∣ = ⌊ \frac{50}{4} ⌋ = 12, ∣ B ∣ = ⌊ \frac{50}{6} ⌋ = 8, ∣ C ∣ = ⌊ \frac{50}{7} ⌋ = 7

∣ A \cap B ∣

lcm (4, 6) = 12 \Rightarrow ⌊ \frac{50}{12} ⌋ = 4

∣ B \cap C ∣

lcm (6, 7) = 42 \Rightarrow ⌊ \frac{50}{42} ⌋ = 1

∣ A \cap C ∣

lcm (4, 7) = 28 \Rightarrow ⌊ \frac{50}{28} ⌋ = 1

∣ A \cap B \cap C ∣

lcm (4, 6, 7) = 84 \Rightarrow ⌊ \frac{50}{84} ⌋ = 0

∣ A \cup B \cup C ∣ = 12 + 8 + 7 - 4 - 1 - 1 + 0 = 21

.

Probability

= \frac{21}{50}

JEE Main · 2024 · 29 Jan 2024 · Shift 2 · Single correct option