Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
Number of ways in which 9 different toys be distributed among 4 children belonging to different age groups in such a way that distribution among the 3 elder children is even and the youngest one is to receive one toy more, is:
\binom{9}{\frac{(5)}{2}}
\frac{9!}{2}
\frac{9!}{3!(2)^3}
None of these
Step-by-Step Solution
Key Concept: Binomial coefficient sums with alternating upper or lower indices follow patterns that can be summed using the hockey stick identity.
For the binomial expansion coefficients $\beta_{100}$ and $\beta_{99}$, we write $\beta_{100} = {}^{100}C_{100} + {}^{100}C_{99} + {}^{100}C_{98} + ... + {}^{100}C_0 - {}^{100}C_{100}$ and $\beta_{99} = {}^{99}C_{99} + {}^{100}C_{99} + {}^{101}C_{99} + ... + {}^{200}C_{99} - {}^{201}C_{100}$. These represent specific partial sums of binomial coefficients that can be computed using hockey stick identities and generating function techniques.
Correct Answer: 3