QJEEM19Ma6966: JEE Main 2019 Mathematics - Class 11 JEEM19Ma6966 Solution: If   then K is equal to :

Question

Verified Solution • 2025–26 Session moderate

If

\sum\limits_{r = 0}^{25} {\left\{ {{}^{50}{C_r}.{}^{50 - r}{C_{25 - r}}} \right\} = K\left( {^{50}{C_{25}}} \right)} ,\,\,

then K is equal to :

Accepted Answer

\sum\limits_{r = 0}^{25} {^{50}} {C_r}.{}^{50 - r}{C_{25 - r}}

= \sum\limits_{r = 0}^{25} {{{50!} \over {r!\left( {50 - r} \right)!}}} \times {{\left( {50 - r} \right)!} \over {\left( {25} \right)!\left( {25 - r} \right)!}}

= \sum\limits_{r = 0}^{25} {{{50!} \over {25!25!}} \times {{25!} \over {\left( {25 - r} \right)!\left( {r!} \right)}}}

= {}^{50}{C_{25}}\sum\limits_{r = 0}^{25} {^{25}} {C_r} = \left( {{2^{25}}} \right){}^{50}{C_{25}}

\therefore

K = {2^{25}}

QJEEM19Ma6966: JEE Main 2019 Mathematics - Class 11 JEEM19Ma6966 Solution: If&nbsp;&nbsp; then K is equal to :