class 11

Math

Algebra

Binomial Theorem

The value of (21C1 $−$10C1) $+$(21C2 $−$10C2) $+$(21C3 $−$10C3) $+$(21C4 - 10C4) $++$(21C10 $−$10C10), is $2_{20}−2_{9}$ (2) $2_{20}−2_{10}$ (3) $2_{21}−2_{11}$ (4) $2_{21}−2_{10}$

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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

The expansion $1+x,1+x+x_{2},1+x+x_{2}+x_{3},….1+x+x_{2}+…+x_{20}$ are multipled together and the terms of the product thus obtained are arranged in increasing powers of $x$ in the form of $a_{0}+a_{1}x+a_{2}x_{2}+…$, then, Number of terms in the product