Mathematics · Definite Integration

JEE Main 2026 — 4 April, Evening Shift — Question 45

From a month of 31 days, 3 different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\dfrac{a}{b}$ , where $a,\ b \in \mathbb{N}$ and $\gcd(a,b) = 1$ , then $a + b$ is equal to $\underline{\hspace{1.5cm}}.$

Answer: 944

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Step-by-step solution

Total numbers of ways of selecting 3 numbers $={ }^{31} \mathrm{C}_{3}=\frac{31 \times 30 \times 29}{3 \times 2 \times 1}=4495$ $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in A.P. ⇒ either $\mathrm{a} \& \mathrm{c}$ both odd numbers or both even numbers $={ }^{16} \mathrm{C}_{2}+{ }^{15} \mathrm{C}_{2}$ $=120+105$ $=225$ Probability $=\frac{225}{4495}=\frac{45}{899}$ $=\frac{\mathrm{a}}{\mathrm{b}} \Rightarrow \mathrm{a}+\mathrm{b}=944$

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Exam: JEE Main 2026
Paper: 4 April, Evening Shift
Subject: Mathematics
Chapter: Definite Integration
Topic: Introduction to Definite Integration

Answer: 944

Step-by-step solution

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