Question
Download Solution PDF\(\mathop \sum \limits_{r = 0}^n C\left( {n,r} \right)\) किसके बराबर है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
द्विपद प्रसरण:
\((a+b)^{n}=\sum_{r=0}^{n}\left(\begin{array}{l} n \\ r \end{array}\right) a^{(r)} b^{(n-r)}\)
गणना:
\(\mathop \sum \limits_{r = 0}^n C\left( {n,r} \right)=\left(\begin{array}{l}n \\ 0\end{array}\right)+\left(\begin{array}{l}n \\ 1\end{array}\right)+\left(\begin{array}{l}n \\ 2\end{array}\right)+\cdots+\left(\begin{array}{l}n \\ n\end{array}\right)\)
\((1+x)^{n}=\left(\begin{array}{l}n \\ 0\end{array}\right)(1)^{0}(x)^{n}+\left(\begin{array}{l}n \\ 1\end{array}\right)(1)^{0}(x)\cdots\\ \text{If we put } x=1, \text { we get}\\ (2)^{n}=\left(\begin{array}{l}n \\ 0\end{array}\right)+\left(\begin{array}{l}n \\ 1\end{array}\right)+\left(\begin{array}{l}n \\ 2\end{array}\right)+\cdots+\left(\begin{array}{l}n \\ n\end{array}\right)\)
∴\(\mathop \sum \limits_{r = 0}^n C\left( {n,r} \right)=2^n\)
अतः विकल्प (4) सही है।
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