Probability
Classical Probability
Grade 12
Question:
<p>If 10 different balls are placed in 4 distinct boxes at random, the probability that two of these boxes contain exactly 2 and exactly 3 balls is <em>[JEE Main 2020]</em></p>
<p>\(\dfrac{945}{2^{11}}\)</p>
<p>\(\dfrac{945}{2^{10}}\)</p>
<p>\(\dfrac{965}{2^{11}}\)</p>
<p>\(\dfrac{945}{4^{10}}\)</p>
Step-by-Step Solution
Key Concept: Choose which 2 boxes get 2 and 3 balls. Distribute balls. Divide by total = 4^10.
<p>Total ways: $4^{10}$.</p><p>Favorable: Choose which box gets 2 and which gets 3 ($4\times3=12$ ordered pairs). Choose 2 balls for box A: $\binom{10}{2}$. Choose 3 from remaining 8: $\binom{8}{3}$. Remaining 5 go into the other 2 boxes in $2^5$ ways.</p><p>Favorable $= 12 \times \binom{10}{2}\times\binom{8}{3}\times 2^5 = 12\times45\times56\times32$.</p><p>$= 12\times45\times56\times32 = 12\times45\times1792 = 967680$.</p><p>$P = \dfrac{967680}{4^{10}} = \dfrac{967680}{1048576}$. Simplify: $\frac{945}{2^{10}}$. Answer: B.</p>
Correct Answer: B