A) \[{{2}^{24}}\]
B) \[{{2}^{25}}-1\]
C) \[{{2}^{25}}\]
D) \[{{\left( 25 \right)}^{2}}\]
Correct Answer: C
Solution :
\[\sum\limits_{r=0}^{25}{^{50}{{C}_{r}}\,.{{\,}^{50-r}}{{C}_{25-r}}}\] \[=\,\,\,\sum\limits_{r=0}^{25}{\frac{50!}{r!\left( 50-r \right)!}\,\times \frac{(50-r)}{25!\left( 25-r \right)!}}\] \[=\,\,{{\,}^{50}}{{C}_{25}}\,\,\,\sum\limits_{r=0}^{25}{{{\,}^{25}}{{C}_{r}}={{2}^{25}}\,.{{\,}^{50}}{{C}_{25}}}\] \[\therefore \,\,\,K={{2}^{25}}\]
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