Probability
Sum of Two Positive Integers — Product Constraint
nta_pyq_2024_apr
Grade 12
Question:
Let the sum of two positive integers be 24. If the probability, that their product is not less than $\frac{3}{4}$ times their greatest possible product, is $\frac{m}{n}$, where $\gcd(m,n)=1$, then $n-m$ equals
Step-by-Step Solution
Key Concept: If $x+y=24$, max product is at $x=y=12$: $xy_{\max}=144$. Need $xy\geq\frac{3}{4}\times144=108$. Total positive integer pairs: $(1,23),(2,22),\ldots,(23,1)$, i.e. 23 pairs.
13 favourable pairs out of 23. $p=13/23$. $n-m=10$.
Correct Answer: 1