Mathematics · Probability

JEE Advanced 2019 — Paper 2 — Question 32

Let $|\mathrm{x}|$ denote the number of elements in a set X . Let $\mathrm{S}=\{1,2,3,4,5,6\}$ be a sample space, where each element is equally likely to occur. If $A$ and $B$ are independent events associated with $S$ , then the number of ordered pairs (A, B) such that $1 \leq|B|<|A|$ , equals $\_\_\_\_$

Answer: 422

Numerical answer — enter this value.

Step-by-step solution

Let $|A|=x,|B|=y,|A \cap B|=P$ , then $\frac{P}{6}=\frac{x}{6} \cdot \frac{y}{6}$

$\Rightarrow 6 P=x y$

If $P=1 \Rightarrow$ (i) $x=6, y=1 \Rightarrow{ }^{6} C_{6} \cdot{ }^{6} \mathrm{C}_{1}=6$

or $\Rightarrow$ (ii) $\mathrm{x}=3, \mathrm{y}=2 \Rightarrow{ }^{6} \mathrm{C}_{3} \cdot{ }^{3} \mathrm{C}_{1} \cdot{ }^{3} \mathrm{C}_{1}=180$

If $\mathrm{P}=2 \Rightarrow$ (i) $\mathrm{x}=6, \mathrm{y}=2 \Rightarrow{ }^{6} \mathrm{C}_{6} \cdot{ }^{6} \mathrm{C}_{2}=15$

or (ii) $\mathrm{x}=4, \mathrm{y}=3 \Rightarrow{ }^{6} \mathrm{C}_{4} \cdot{ }^{4} \mathrm{C}_{2} \cdot{ }^{2} \mathrm{C}_{1}=180$

If $P=3 \Rightarrow x=6, y=3 \Rightarrow{ }^{6} \mathrm{C}_{6} \cdot{ }^{6} \mathrm{C}_{3}=20$

If $\mathrm{P}=4 \Rightarrow \mathrm{x}=6, \mathrm{y}=4 \Rightarrow{ }^{6} \mathrm{C}_{6} \cdot{ }^{6} \mathrm{C}_{4}=15$

If $\mathrm{P}=5 \Rightarrow \mathrm{x}=6, \mathrm{y}=5 \Rightarrow{ }^{6} \mathrm{C}_{6} \cdot{ }^{6} \mathrm{C}_{5}=6$

Hence total number of ordered pairs $(A, B)=422$ .

Answer key and solution verified before publishing.

Practise Probability

Start with this question, then two more from the same chapter — with a tutor that explains every step. Free.

Solve a similar one free

Exam: JEE Advanced 2019
Paper: Paper 2
Subject: Mathematics
Chapter: Probability
Topic: Independent Events