JEE Advanced 2019 · Paper 2 · Mathematics Q11

← JEE Advanced 2019 · Paper 2

MathematicsNumerical+3Question 47 of 54

Let $|X|$ denote the number of elements in a set $X$ . Let $S = \{1, 2, 3, 4, 5, 6\}$ 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 $1 \le |B| < |A|$ , equals ______

Your answer

Answer and worked solution

Official answer: 422

Independence. $P(A\cap B)=P(A)P(B)$ means $6|A\cap B|=|A||B|$ , so $|A||B|$ must be a multiple of 6. With $1\le|B|<|A|\le6$ :

$|A|=6$ , any $|B|$ from 1 to 5: $\sum_{b=1}^5\binom6b=62$ .
$|A|=3$ , $|B|=2$ , with $|A\cap B|=1$ : $\binom63\cdot3\cdot3=180$ .
$|A|=4$ , $|B|=3$ , with $|A\cap B|=2$ : $\binom64\cdot\binom42\cdot2=180$ .

No other pair of sizes gives a product divisible by 6.

Total. $62+180+180=422$ .

Answer: 422

Topics:Independent events Combinations

← Previous question Next question →

JEE Advanced 2019 · Paper 2, Mathematics question 11