${(1 + x + {x^2})^n}$ के विस्तार में गुणांकों का योग होगा
$2$
${3^n}$
${4^n}$
${2^n}$
गुणांकों के योगफल के लिए $x = 1$ रखने पर,
==> ${(1 + x + {x^2})^n} = {(1 + 1 + 1)^n} = {3^n}$.
$2{C_0} + \frac{{{2^2}}}{2}{C_1} + \frac{{{2^3}}}{3}{C_2} + …. + \frac{{{2^{11}}}}{{11}}{C_{10}}$=
यदि $\sum_{ r =0}^{25}\left\{{ }^{50} C _{ r } \cdot{ }^{50- r } C _{25- r }\right\}= K \left({ }^{50} C _{25}\right)$ हो, तो $K$ का मान होगा
यदि ${a_k} = \frac{1}{{k(k + 1)}},$ जबकि $k = 1,\,2,\,3,\,4,…..,\,n$, तब ${\left( {\sum\limits_{k = 1}^n {{a_k}} } \right)^2} = $
$\left( {\left( {\begin{array}{*{20}{c}} {21}\\ 1 \end{array}} \right) – \left( {\begin{array}{*{20}{c}} {10}\\ 1 \end{array}} \right)} \right) + \left( {\left( {\begin{array}{*{20}{c}} {21}\\ 2 \end{array}} \right) – \left( {\begin{array}{*{20}{c}} {10}\\ 2 \end{array}} \right)} \right)$$ + \left( {\left( {\begin{array}{*{20}{c}} {21}\\ 3 \end{array}} \right) – \left( {\begin{array}{*{20}{c}} {10}\\ 3 \end{array}} \right)} \right) + \;.\;.\;.$$ + \left( {\left( {\begin{array}{*{20}{c}} {21}\\ {10} \end{array}} \right) – \left( {\begin{array}{*{20}{c}} {10}\\ {10} \end{array}} \right)} \right)$ का मान है:
यदि $1+\left(2+{ }^{49} C _1+{ }^{49} C _2+\ldots \ldots+{ }^{49} C _{49}\right)\left({ }^{50} C _2+\right.$ $\left.{ }^{50} C _4+\ldots . .+{ }^{50} C _{50}\right)=2^{ n } . m$ है, जहाँ $m$ एक विषम संख्या है, तो $n + m$ बराबर है $……….$
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