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Question #41820
The Question
Let be the greatest binomial coefficient in the expansion of . A bag contains distinct balls numbered to . Experiment A consists of drawing 3 balls from the bag with replacement. Let be the probability that the sum of the numbers on the drawn balls is at most . Experiment B consists of drawing 3 balls from the bag without replacement. Let be the probability that the sum of the numbers on the drawn balls is at most . Find the value of .
Options
- A.
- B.
- C.
- D.
Check Your Answer
Correct Answer
Step-by-Step Solution
Explanation
How the Examiner Thinks
Bridge
Practice problems that bridge the gap between foundational theory and advanced practical application.
Concepts Behind This Question
Sampling With/Without Replacement|Selection vs Arrangement|Greatest Binomial Coefficient|Complementary Counting
Why This Question Is Tricky
Common Mistakes
This question engineers a collision between multinomial distributions (sampling with replacement) and hypergeometric constraints (sampling without replacement).
The boundary condition N=20 is derived from the greatest binomial coefficient.
Students often fail to correctly count the number of distinct selections for a bounded sum.
The trap lies in manually counting or ignoring the edge-case exclusions of identical draws.
By applying Complementary Counting via the Principle of Inclusion-Exclusion (or generating functions) on the unrestricted integer partitions, we elegantly bypass redundant calculations.
Options include traps where students use combinations with replacement for the sample space or fail to divide by 3! when removing permutations of identical draws.
Quick Facts About This Question
Question Information
| Subject | Mathematics |
| Chapter | Probability - Class 12 |
| Concept | Sampling With/Without Replacement |
| Difficulty | 8 |
| Cognitive Load | 7 |
| Target Solve Time | 160 sec |
| Access | Free |