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7.Binomial Theorem
hard
$(1-x)^{100}$ ના દ્વિપદી વિસ્તરણમાં પ્રથમ $50$ પદોના સહગુણકોનો સરવાળો $.......$ છે.
A
$-{ }^{101} C _{50}$
B
${ }^{99} C _{49}$
C
$-{ }^{99} C _{49}$
D
${ }^{101} C _{50}$
(JEE MAIN-2023)
Solution
$(1- x )^{100}= Co – C _1 x + C _2 x ^2-$
$C _3 x ^3+\ldots C _{99 x }{ }^{99}+ C _{100 x } x ^{10}$
$\Rightarrow Co ^{- C _1}+ C _2- C _3+\ldots \ldots- C _{99}+ C _{100}=0$
$2\left( Co – C _1+ C _2+\ldots \ldots- C _9\right)+ C _{50}=0$
$C _0- C _1+ C _2+\ldots . C _{99}=-\frac{1}{2}{ }^{100} C _{50}$
$-\frac{1}{2} \frac{100 !}{50 ! 50 !}=-\frac{1}{2} \times \frac{100 \times 99 !}{50 ! 50 !}=-{ }^{99} C _{49}$
Standard 11
Mathematics
