Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
There are 10 seats in a double decker bus, 6 in the lower deck and 4 on the upper deck. Ten passenger board the bus, of them 3 refuse to go to the upper deck and 2 insist on going up. The number of way in which the passengers can be accommodated is: (Assume all seats to be duly numbered)
172800
162800
152800
182800
Step-by-Step Solution
Key Concept: Binomial coefficient sums can be bounded using arithmetic-geometric mean inequalities applied to specific groupings.
Using the constraint that $^nC_1 + ^nC_2 + ... + ^nC_{n-1}$ is less than the arithmetic mean of coefficients, we establish an inequality. The problem applies the AM inequality to binomial coefficients to derive $p < \left(\frac{2^n - 2}{n-1}\right)^{n-1}$ and also $p < \left(\frac{2^n}{n+1}\right)^{n+1}$ using different groupings of coefficients.
Correct Answer: 1