Class 12

Math

Algebra

Binomial Theorem

A path of length $n$ is a sequence of points $(x_{1},y_{1})$, $(x_{2},y_{2})$,….,$(x_{n},y_{n})$ with integer coordinates such that for all $i$ between $1$ and $n−1$ both inclusive,

either $x_{i+1}=x_{i}+1$ and $y_{i+1}=y_{i}$ (in which case we say the $i_{th}$ step is rightward)

or $x_{i+1}=x_{i}$ and $y_{i+1}=y_{i}+1$ ( in which case we say that the $i_{th}$ step is upward ).

This path is said to start at $(x_{1},y_{1})$ and end at $(x_{n},y_{n})$. Let $P(a,b)$, for $a$ and $b$ non-negative integers, denotes the number of paths that start at $(0,0)$ and end at $(a,b)$.

Number of ordered pairs $(i,j)$ where $i=j$ for which $P(i,100−i)=P(i,100−j)$ is