Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade None
Question:
The combinatorial $C(n, r)$ is equal to:
number of possible subsets of r members from a set of n distinct members
number of possible binary messages of length n with exactly r 1's
number of non decreasing 2-D paths from the lattice point (0,0) to (r, n)
number of ways of selecting r things out of n different things when a particular thing is always included plus the number of ways of selecting 'r' things out of n, when a particular thing is always excluded
Step-by-Step Solution
Key Concept: Matching corresponding terms in two binomial expansions determines the index of summation and allows computing the sum.
The coefficient of $x^j$ in the expansion $(x + c_1)(x + c_2)...(x + c_n)$ can be found by comparing expansions. Given two different coefficient expressions, we set the indices equal: $n - t = t + 2$, giving $t = \frac{n-2}{2}$. The sum of these coefficients is ${}^{n+2}C_{t+2} = {}^{n+2}C_{\frac{n+2}{2}}$.
Correct Answer: 1,2,4