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Simplify binom{100}{50}/binom{200}{51}*binom{100}{51}/binom{200}{50}

 Dec 3, 2022
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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

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