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 #1
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\(\frac{\binom{100}{50}}{\binom{200}{51}}*\frac{\binom{100}{51}}{\binom{200}{50}}\\ =\binom{100}{50}*{\binom{100}{51}} \div \left[ {\binom{200}{51}}* {\binom{200}{50}} \right]\\ =\frac{100!*100!}{50!*50!*51!*49!} \div \left[ \frac{200!*200!}{51!*149!*50!*150!}\right]\\ =\frac{100!*100!}{50!*50!*51!*49!} \times \left[ \frac{51!*149!*50!*150!}{200!*200!}\right]\\ =\frac{100!*100!}{50!*49!} \times \left[ \frac{149!*150!}{200!*200!}\right]\\ =\frac{50*51^2*52^2* \dots 100^2}{1} \times \left[ \frac{1}{150*151^2*152^2*\dots 200^2}\right]\\ =\frac{51^2*52^2* \dots 100^2}{1} \times \left[ \frac{1}{3*(151^2*153^2*155^2*\dots 199^2)*(152^2*154^2*\dots 200^2}\right]\\ =\frac{51^2*52^2* \dots 100^2}{1} \times \left[ \frac{1}{3*(151^2*153^2*155^2*\dots 199^2)*4^{25}(76^2*77^2*\dots 100^2)}\right]\\ =\frac{51^2*52^2* \dots 75^2}{1} \times \left[ \frac{1}{3*4^{25}*(151^2*(3*51)^2*155^2*157^2*(3*53)^2*\dots*(3*63)^2* 199^2)}\right]\\ \)

 

 

\(=\frac{51^2*52^2*  \dots   75^2}{1} \times \left[ \frac{1}{3*4^{25}*(151^2*(3*51)^2*155^2*157^2*(3*53)^2*\dots*(3*63)^2* 199^2)}\right]\\ =\frac{51^2*52^2*  \dots   75^2}{1} \times \left[ \frac{1}{3*4^{25}*(151^2\;*\;155^2*157^2\:*\:161^2*\dots** 199^2)*3^7*(51^2*53^2*\dots 63^2)}\right]\\ =\frac{(52^2*54^2\dots 62^2)  \dots 64^2*65^2\dots   75^2}{1} \times \left[ \frac{1}{3*4^{25}*(151^2\;*\;155^2*157^2\:*\:161^2*\dots** 199^2)*3^7}\right]\\ =\frac{(52^2*54^2\dots 62^2)  \dots 64^2*65^2\dots   75^2}{1} \times \left[ \frac{1}{3^8*4^{25}*(151^2\;*\;155^2*157^2\:*\:161^2*\dots** 199^2)}\right]\\\)

 

etc

 

 

 

 

 

 

 

LaTex:

\frac{\binom{100}{50}}{\binom{200}{51}}*\frac{\binom{100}{51}}{\binom{200}{50}}\\

=\binom{100}{50}*{\binom{100}{51}} \div \left[ {\binom{200}{51}}*  {\binom{200}{50}}            \right]\\

=\frac{100!*100!}{50!*50!*51!*49!} \div \left[ \frac{200!*200!}{51!*149!*50!*150!}\right]\\

=\frac{100!*100!}{50!*50!*51!*49!} \times \left[ \frac{51!*149!*50!*150!}{200!*200!}\right]\\

=\frac{100!*100!}{50!*49!} \times \left[ \frac{149!*150!}{200!*200!}\right]\\

=\frac{50*51^2*52^2*  \dots   100^2}{1} \times \left[ \frac{1}{150*151^2*152^2*\dots 200^2}\right]\\
=\frac{51^2*52^2*  \dots   100^2}{1} \times \left[ \frac{1}{3*(151^2*153^2*155^2*\dots 199^2)*(152^2*154^2*\dots 200^2}\right]\\
=\frac{51^2*52^2*  \dots   100^2}{1} \times \left[ \frac{1}{3*(151^2*153^2*155^2*\dots 199^2)*4^{25}(76^2*77^2*\dots 100^2)}\right]\\

=\frac{51^2*52^2*  \dots   75^2}{1} \times \left[ \frac{1}{3*4^{25}*(151^2*(3*51)^2*155^2*157^2*(3*53)^2*\dots*(3*63)^2* 199^2)}\right]\\

 

=\frac{51^2*52^2*  \dots   75^2}{1} \times \left[ \frac{1}{3*4^{25}*(151^2*(3*51)^2*155^2*157^2*(3*53)^2*\dots*(3*63)^2* 199^2)}\right]\\
=\frac{51^2*52^2*  \dots   75^2}{1} \times \left[ \frac{1}{3*4^{25}*(151^2\;*\;155^2*157^2\:*\:161^2*\dots** 199^2)*3^7*(51^2*53^2*\dots 63^2)}\right]\\
=\frac{(52^2*54^2\dots 62^2)  \dots 64^2*65^2\dots   75^2}{1} \times \left[ \frac{1}{3*4^{25}*(151^2\;*\;155^2*157^2\:*\:161^2*\dots** 199^2)*3^7}\right]\\
=\frac{(52^2*54^2\dots 62^2)  \dots 64^2*65^2\dots   75^2}{1} \times \left[ \frac{1}{3^8*4^{25}*(151^2\;*\;155^2*157^2\:*\:161^2*\dots** 199^2)}\right]\\

Dec 3, 2022, 6:02:48 PM
 #3
avatar+118139 
+2

\(\displaystyle (e^\frac{5\pi}{6} + e^\frac{4\pi}{3})^9\\ =\displaystyle \sum_{n=0}^9 \binom{9}{n}(e^\frac{5\pi\; n}{6}) (e^\frac{8\pi (9-n)}{6})\quad \\ = \binom{9}{0}(e^\frac{5\pi\; 0}{6}) (e^\frac{8\pi (9)}{6}) + \binom{9}{1}(e^\frac{5\pi\; 1}{6}) (e^\frac{8\pi (8)}{6}) + \binom{9}{2}(e^\frac{5\pi\; 2}{6}) (e^\frac{8\pi (7)}{6}) + \binom{9}{3}(e^\frac{5\pi\; 3}{6}) (e^\frac{8\pi (6)}{6}) + \binom{9}{4}(e^\frac{5\pi\; 4}{6}) (e^\frac{8\pi (5)}{6}) + \binom{9}{5}(e^\frac{5\pi\; 5}{6}) (e^\frac{8\pi (4)}{6}) + \binom{9}{6}(e^\frac{5\pi\; 6}{6}) (e^\frac{8\pi (3)}{6}) + \binom{9}{7}(e^\frac{5\pi\; 7}{6}) (e^\frac{8\pi (2)}{6}) + \binom{9}{8}(e^\frac{5\pi\; 8}{6}) (e^\frac{8\pi (1)}{6}) + \binom{9}{9}(e^\frac{5\pi\; 9}{6}) (e^\frac{8\pi (0)}{6})\\ = (e^\frac{72\pi }{6}) + 9(e^\frac{5\pi}{6}) (e^\frac{64\pi }{6}) + \binom{9}{2}(e^\frac{10\pi}{6}) (e^\frac{56\pi }{6}) + \binom{9}{3}(e^\frac{15\pi}{6}) (e^\frac{48\pi }{6}) + \binom{9}{4}(e^\frac{20\pi}{6}) (e^\frac{40\pi }{6}) + \binom{9}{5}(e^\frac{25\pi }{6}) (e^\frac{32\pi }{6}) + \binom{9}{6}(e^\frac{30\pi}{6}) (e^\frac{24\pi }{6}) + \binom{9}{7}(e^\frac{35\pi}{6}) (e^\frac{16\pi }{6}) +9 (e^\frac{40\pi}{6}) (e^\frac{8\pi }{6}) + (e^\frac{45\pi}{6}) \\ = (e^\frac{72\pi }{6}) + 9 (e^\frac{69\pi }{6}) + \binom{9}{2}(e^\frac{66\pi}{6}) + \binom{9}{3}(e^\frac{63\pi}{6}) + \binom{9}{4}(e^\frac{60\pi}{6}) + \binom{9}{5}(e^\frac{57\pi }{6}) + \binom{9}{6}(e^\frac{54\pi}{6}) + \binom{9}{7}(e^\frac{51\pi}{6}) +9 (e^\frac{48\pi}{6}) + (e^\frac{45\pi}{6}) \\ = (e^\frac{24\pi }{2}) + 9 (e^\frac{23\pi }{2}) + \binom{9}{2}(e^\frac{22\pi}{2}) + \binom{9}{3}(e^\frac{21\pi}{2}) + \binom{9}{4}(e^\frac{20\pi}{2}) + \binom{9}{5}(e^\frac{19\pi }{2}) + \binom{9}{6}(e^\frac{18\pi}{2}) + \binom{9}{7}(e^\frac{17\pi}{2}) +9 (e^\frac{16\pi}{2}) + (e^\frac{15\pi}{2}) \\\)

 

 

LaTex

\displaystyle (e^\frac{5\pi}{6} + e^\frac{4\pi}{3})^9\\
=\displaystyle \sum_{n=0}^9    \binom{9}{n}(e^\frac{5\pi\; n}{6})  (e^\frac{8\pi (9-n)}{6})\quad \\
= \binom{9}{0}(e^\frac{5\pi\; 0}{6})  (e^\frac{8\pi (9)}{6})
+ \binom{9}{1}(e^\frac{5\pi\; 1}{6})  (e^\frac{8\pi (8)}{6})
+ \binom{9}{2}(e^\frac{5\pi\; 2}{6})  (e^\frac{8\pi (7)}{6})
+ \binom{9}{3}(e^\frac{5\pi\; 3}{6})  (e^\frac{8\pi (6)}{6})
+ \binom{9}{4}(e^\frac{5\pi\; 4}{6})  (e^\frac{8\pi (5)}{6})
+ \binom{9}{5}(e^\frac{5\pi\; 5}{6})  (e^\frac{8\pi (4)}{6})
+ \binom{9}{6}(e^\frac{5\pi\; 6}{6})  (e^\frac{8\pi (3)}{6})
+ \binom{9}{7}(e^\frac{5\pi\; 7}{6})  (e^\frac{8\pi (2)}{6})
+ \binom{9}{8}(e^\frac{5\pi\; 8}{6})  (e^\frac{8\pi (1)}{6})
+ \binom{9}{9}(e^\frac{5\pi\; 9}{6})  (e^\frac{8\pi (0)}{6})\\

= (e^\frac{72\pi }{6})
+ 9(e^\frac{5\pi}{6})  (e^\frac{64\pi }{6})
+ \binom{9}{2}(e^\frac{10\pi}{6})  (e^\frac{56\pi }{6})
+ \binom{9}{3}(e^\frac{15\pi}{6})  (e^\frac{48\pi }{6})
+ \binom{9}{4}(e^\frac{20\pi}{6})  (e^\frac{40\pi }{6})
+ \binom{9}{5}(e^\frac{25\pi }{6})  (e^\frac{32\pi }{6})
+ \binom{9}{6}(e^\frac{30\pi}{6})  (e^\frac{24\pi }{6})
+ \binom{9}{7}(e^\frac{35\pi}{6})  (e^\frac{16\pi }{6})
+9 (e^\frac{40\pi}{6})  (e^\frac{8\pi }{6})
+ (e^\frac{45\pi}{6}) \\

= (e^\frac{72\pi }{6})
+ 9 (e^\frac{69\pi }{6})
+ \binom{9}{2}(e^\frac{66\pi}{6})  
+ \binom{9}{3}(e^\frac{63\pi}{6})  
+ \binom{9}{4}(e^\frac{60\pi}{6})  
+ \binom{9}{5}(e^\frac{57\pi }{6})  
+ \binom{9}{6}(e^\frac{54\pi}{6})  
+ \binom{9}{7}(e^\frac{51\pi}{6})  
+9 (e^\frac{48\pi}{6}) 
+ (e^\frac{45\pi}{6}) \\

Nov 23, 2022