Probability
Inclusion-Exclusion — Multiples in a Set
nta_pyq_2024_jan
Grade 12
Question:
An integer is chosen at random from the integers $1,2,3,\ldots,50$. The probability that the chosen integer is a multiple of at least one of $4$, $6$ and $7$ is
$\dfrac{8}{25}$
$\dfrac{21}{50}$
$\dfrac{9}{50}$
$\dfrac{14}{25}$
Step-by-Step Solution
Key Concept: Apply inclusion-exclusion. Count multiples of 4, 6, 7 in $\{1,\ldots,50\}$: $|A|=12$, $|B|=8$, $|C|=7$. Intersections: $|A\cap B|=|\text{lcm}(4,6)|=|12|=4$, $|B\cap C|=|42|=1$, $|A\cap C|=|28|=1$, $|A\cap B\cap C|=|\text{lcm}(4,6,7)|=|84|=0$.
$12+8+7-4-1-1+0=21$. $P=21/50$.
Correct Answer: 2