$\frac{1}{{1!(n - 1)\,!}} + \frac{1}{{3!(n - 3)!}} + \frac{1}{{5!(n - 5)!}} + .... = $

  • A

    $\frac{{{2^n}}}{{n!}}$; for all even values of $n$

  • B

    $\frac{{{2^{n - 1}}}}{{n!}}$; for all values of $n$ i.e., all even odd values

  • C

    $0$

  • D

    None of these

Similar Questions

For natural numbers $m,n$ ,if ${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n} = 1 + {a_1}y + {a_2}{y^2} + \ldots \;$ and $a_1= a_2=10,$ then $(m,n)$ =______. 

  • [AIEEE 2006]

If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .......... + {C_n}{x^2},$ then $C_0^2 + C_1^2 + C_2^2 + C_3^2 + ...... + C_n^2$ =

In the expansion of ${(1 + x)^5}$, the sum of the coefficient of the terms is

If $\sum_{ k =1}^{10} K ^{2}\left(10_{ C _{ K }}\right)^{2}=22000 L$, then $L$ is equal to $.....$

  • [JEE MAIN 2022]

If the sum of the coefficients in the expansion of ${({\alpha ^2}{x^2} - 2\alpha {\rm{ }}x + 1)^{51}}$ vanishes, then the value of $\alpha $ is

  • [IIT 1991]