Probability — Expectation of a Discrete Random Variable
PYP_JEE_ADV_2026_P2
Grade None
Question:
A bookshelf contains 6 distinct books of Mathematics and 5 distinct books of Physics. From these 11 books, 6 books are chosen at random. Let $X$ be the absolute value of the difference between the number of Mathematics books chosen and the number of Physics books chosen. If $\alpha$ is the mean of the random variable $X$, then the value of $77\alpha$ is __________.
Step-by-Step Solution
Key Concept: Systematically enumerate all possible splits $(m,p)$ with $m+p=6$, compute $X=|m-p|$ for each, and weight by the number of ways to choose that split.
**Step 1: Set up distribution of X**
If $m$ Math and $p=6-m$ Physics books chosen: $X=|m-p|=|2m-6|$. Total ways $=\binom{11}{6}=462$.
**Step 2: Compute probabilities**
| $m$ | $X$ | Ways $\binom{6}{m}\binom{5}{6-m}$ |
|---|---|---|
|1|4|$6\times1=6$|
|2|2|$15\times5=75$|
|3|0|$20\times10=200$|
|4|2|$15\times10=150$|
|5|4|$6\times5=30$|
|6|6|$1\times1=1$|
**Step 3: Compute $E(X)$**
$\alpha=\frac{4\times6+2\times75+0\times200+2\times150+4\times30+6\times1}{462}=\frac{24+150+0+300+120+6}{462}=\frac{600}{462}=\frac{100}{77}$. $77\alpha=100$.
Correct Answer: 100