ધારો કે $\alpha=\sum_{r=0}^n\left(4 r^2+2 r+1\right)^n C_r$ અને $\beta=\left(\sum_{r=0}^n \frac{{ }^n C_r}{r+1}\right)+\frac{1}{n+1} \cdot$ જો $140 < \frac{2 \alpha}{\beta}<281$ તો $n$ નું મૂલ્ય .......... છે.
$9$
$4$
$5$
$6$
જો ${({\alpha ^2}{x^2} - 2\alpha {\rm{ }}x + 1)^{51}}$ ના સહગુણકનો સરવાળો શૂન્ય હોય તો $\alpha $ મેળવો.
જો ${C_r}$ એ $^n{C_r}$ દર્શાવે છે તો , $\frac{{2(n/2)!(n/2)!}}{{n!}}[C_0^2 - 2C_1^2 + 3C_2^2 - ..... + {( - 1)^n}(n + 1)C_n^2]$ મેળવો. (કે જ્યાં $n$ એ યુગ્મ પુર્ણાક છે )
$\frac{{{C_0}}}{1} + \frac{{{C_2}}}{3} + \frac{{{C_4}}}{5} + \frac{{{C_6}}}{7} + ....$=
જો ${S_n} = \sum\limits_{r = 0}^n {\frac{1}{{^n{C_r}}}} $ અને ${t_n} = \sum\limits_{r = 0}^n {\frac{r}{{^n{C_r}}}} $, તો $\frac{{{t_n}}}{{{S_n}}}$ = . . .
$\left( \begin{array}{l}30\\0\end{array} \right)\,\left( \begin{array}{l}30\\10\end{array} \right) - \left( \begin{array}{l}30\\1\end{array} \right)\,\left( \begin{array}{l}30\\11\end{array} \right)$ + $\left( \begin{array}{l}30\\2\end{array} \right)\,\left( \begin{array}{l}30\\12\end{array} \right) + ....... + \left( \begin{array}{l}30\\20\end{array} \right)\,\left( \begin{array}{l}30\\30\end{array} \right) = .$ . ..