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$\sum\limits_{i = 0}^m {\left( {\begin{array}{*{20}{c}}{10}\\i\end{array}}\right)} \,\left( {\begin{array}{*{20}{c}}{20}\\{m - i}\end{array}}\right)$,$\left({tcfd\,\left({\begin{array}{*{20}{c}}p\\q\end{array}} \right)\, = 0\,\,;fn\,\,p < q} \right)$ का योग होगा
$5$
$15$
$10$
$20$
Solution
$m =5$ के लिए,$\,\,\,\,\sum\limits_{i = 0}^5 {\left(\begin{array}{l}\,10\,\\\,\,i\end{array} \right)\,\left( \begin{array}{l}\,\,20\,\\5 – i\,\end{array} \right)} $
$ = \left( \begin{array}{l}\,10\\\,\,0\end{array} \right)\,\,\left( \begin{array}{l}20\\\,5\end{array} \right)\,\, + \,\left( \begin{array}{l}10\\\,1\end{array} \right)\,\left( \begin{array}{l}20\\\,4\end{array} \right)\, + … + \left( \begin{array}{l}10\\\,5\end{array} \right)\,\left( \begin{array}{l}20\\\,0\end{array} \right)$ $m = 10$ के लिए, $\sum\limits_{i = 0}^{10} {\left(\begin{array}{l}\,10\,\\\,\,i\end{array} \right)\,\left( \begin{array}{l}\,\,20\,\\10 – i\,\end{array} \right)} $
$ = \left( \begin{array}{l}10\\\,0\end{array} \right)\,\left( \begin{array}{l}20\\10\end{array} \right) + \left( \begin{array}{l}10\\\,1\end{array} \right)\,\left( \begin{array}{l}20\\\,9\,\end{array} \right) + \left( \begin{array}{l}10\\\,2\end{array} \right)\,\left( \begin{array}{l}20\\\,8\,\end{array} \right)$$ + … + \left( \begin{array}{l}10\,\\10\end{array} \right)\,\left( \begin{array}{l}20\\\,0\end{array} \right)$
$m = 15$ के लिए, $\sum\limits_{i = 0}^{15} {\left( \begin{array}{l}\,10\,\\\,\,i\end{array} \right)\,\left( \begin{array}{l}\,\,\,20\,\\15 – i\,\end{array} \right)} $
$ = \left( \begin{array}{l}\,10\\\,\,0\end{array} \right)\,\,\left( \begin{array}{l}20\\\,15\end{array} \right)\,\, + \,\left( \begin{array}{l}10\\\,1\end{array} \right)\,\left( \begin{array}{l}20\\1\,4\end{array} \right)\, + \left( \begin{array}{l}10\\\,2\end{array} \right)\,\left( \begin{array}{l}20\\\,13\end{array} \right) + .. + \left( {\begin{array}{*{20}{c}}{10}\\{10}\end{array}} \right)\,\left( {\begin{array}{*{20}{c}}{20}\\5\end{array}} \right)$
$m = 20$ के लिए, $\sum\limits_{i = 0}^{20} {\left( \begin{array}{l}\,10\,\\\,\,i\end{array} \right)\,\left( \begin{array}{l}\,\,\,20\,\\20 – i\,\end{array} \right)} $
$ = \left( \begin{array}{l}\,10\\\,\,0\end{array} \right)\,\,\left( \begin{array}{l}20\\20\end{array} \right)\,\, + \,\left( \begin{array}{l}10\\\,1\end{array} \right)\,\left( \begin{array}{l}20\\1\,9\end{array} \right)\, + … + \left( \begin{array}{l}10\\10\end{array} \right)\,\left( \begin{array}{l}20\\\,10\end{array} \right)$
स्पष्टत:,$ m = 15$ के लिए योग अधिकतम है।
$r = 5$ के लिए $^{10}{C_r}$ अधिकतम है, तथा $r = 10$ के लिए $^{20}{C_r}$ अधिकतम है। ध्यान दें कि $m = 15$ के लिए एकल पद $^{10}{C_5}\, \times {\,^{20}}{C_{10}}\,$का मान, योगफल
$^{10}{C_0}{\,^{20}}{C_{10}}\, + {\,^{10}}{C_1}{\,^{20}}{C_9}\, + {\,^{10}}{C_2}{\,^{20}}{C_8}\, + …..$
${ + ^{10}}{C_8}{\,^{20}}{C_2}\, + {\,^{10}}{C_9}{\,^{20}}{C_1}\, + {\,^{10}}{C_{10}}{\,^{20}}{C_0}$ ($m=10$ के लिये) से अधिक है।
साथ ही, $m = 10$ के लिए तथा $m = 20$ के लिए योग एक समान है।
