Binomial Theorem Ques 1
- For any positive integers $m, n$ (with $n \geq m$ ), If $\binom{n}{m}={ }^n C_m$. Prove that
$ \binom{n}{m}+\binom{n-1}{m}+\binom{n-2}{m}+\ldots+\binom{m}{m}=\binom{n+1}{m+1} $
or
Prove that
$ \begin{gathered} \binom{n}{m}+2\binom{n-1}{m}+3\binom{n-2}{m}+\ldots+(n-m+1) \binom{m}{m}=\binom{n+2}{m+2} \end{gathered} $
(IIT JEE 2000, 6M)
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Solution: Let $S=\binom{n}{m}+\binom{n-1}{m}+\binom{n-2}{m}+\ldots+\binom{m}{m}=\binom{n+1}{m+1} \quad ……..(i)$
It is obvious that, $n \geq m \quad $ [given]
Note: This question is based upon additive loop.
Now,
$ \begin{aligned} & S=\binom{m}{m}+\binom{m+1}{m}+\binom{m+2}{m}+\ldots+\binom{n}{m} \\ & =\left\{\binom{m+1}{m+1}+\binom{m+1}{m}\right\}+\binom{m+2}{m}+\ldots\binom{n}{m} \\ & {\left[\because\binom{m}{m}=1=\binom{m+1}{m+1}\right]} \\ & =\binom{m+2}{m+1}+\binom{m+2}{m}+\ldots+\binom{n}{m} \\ & {\left[:{ }^n C_r+{ }^n C_{r+1}={ }^{n+1} C_{r+1}\right]} \\ & =\binom{m+3}{m+1}+\ldots+\binom{n}{m} \\ & =\text {……………………………… } \\ & =\binom{n}{m+1}+\binom{n}{m}=\binom{n+1}{m+1} \text {, which is true. } \quad ……..(ii) \\ \end{aligned} $
Again, we have to prove that
$ \binom{n}{m}+2\binom{n-1}{m}+3\binom{n-2}{m}+\ldots+(n-m+1)\binom{m}{m}=\binom{n+2}{m+2} $
Let $S_1=\binom{n}{m}+2\binom{n-1}{m}+3\binom{n-2}{m}+\ldots+(n-m+1)\binom{m}{m}$
$ =\binom{n}{m}+\binom{n-1}{m}+\binom{n-2}{m}+\ldots+\binom{m}{m} $ $ +\binom{n-1}{m}+\binom{n-2}{m}+$
$\ldots+\binom{m}{m} $ $+\binom{n-2}{m}+\ldots+\binom{m}{m} $ $+\binom{m}{m}$
Now, sum of the first row is $\binom{n+1}{m+1}$.
Sum of the second row is $\binom{n}{m+1}$.
Sum of the third row is $\binom{n-1}{m+1}$,
Sum of the last row is $\binom{m}{m}=\binom{m+1}{m+1}$.
Thus,
$S=\binom{n+1}{m+1}+\binom{n}{m+1}+\binom{n-1}{m+1} $ $+\ldots+\binom{m+1}{m+1}=\binom{n+1+1}{m+2}=\binom{n+2}{m+2}$
[from Eq. (i) replacing $n$ by $n+1$ and $m$ by $m+1$ ]
BETA
