Permutation and Combination
Permutation and Combination
Allen Star Batch
Grade 11
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