Frac{1}{1 ! 50 !}+frac{1}{3 ! 48 !}+frac{1}{5 ! 46 !}+ldots .+frac{1}{49 ! 2 !}+frac{1}{51 ! 1 !} ?

English

Gujarati

Hindi

7.Binomial Theorem

hard

$\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}$ ની કિમંત મેળવો.

A

$\frac{2^{50}}{50 !}$

B

$\frac{2^{50}}{51 !}$

C

$\frac{2^{51}}{51 !}$

D

$\frac{2^{51}}{50 \text { ! }}$

(JEE MAIN-2023)

Solution

$\sum \limits_{ r =1}^{26} \frac{1}{(2 r -1) !(51-(2 r -1)) !}=\sum \limits_{ r =1}^{26}{ }^{51} C _{(2 r -1)} \frac{1}{51 !}$

$=\frac{1}{51 !}\left\{{ }^{51} C _1+{ }^{51} C _3+\ldots .+{ }^{51} C _{51}\right\}$

$=\frac{1}{51 !}\left(2^{50}\right)$

Standard 11

Mathematics

Share

Similar Questions

જો $\left(1+x+2 x^{2}\right)^{20}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{40} x^{40}$ હોય તો $a _{1}+ a _{3}+ a _{5}+\ldots+ a _{37}$ ની કિમંત મેળવો.

hard

(JEE MAIN-2021)

$(x – 1)$$\left( {x\, – \,\frac{1}{2}\,} \right)$$\left( {x\, – \,\frac{1}{{{2^2}}}\,} \right)$ …..$\left( {x\, – \,\frac{1}{{{2^{49}}}}\,} \right)$ ના વિસ્તરણમાં $x^{49}$ નો સહગુણક મેળવો

normal

જો ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + …. + {C_n}{x^n}$, તો ${C_0}{C_2} + {C_1}{C_3} + {C_2}{C_4} + {C_{n – 2}}{C_n}$= . . .

medium

$4 \{^nC_1 + 4 . ^nC_2 + 4^2 . ^nC_3 + …… + 4^{n – 1}\}$ ની કિમત મેળવો

normal

${({x^2} – x – 1)^{99}}$ ના સહગુણકનો સરવાળો મેળવો.

easy