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binomial theoram — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSbinomial theoram1 Mark
MCQ
$\sum\limits_{n = 0}^4 {{{\left( {1009 - 2n} \right)}^4}\left( \begin{gathered}
  4 \hfill \\
  n \hfill \\ 
\end{gathered}  \right)} {\left( { - 1} \right)^n}$   is
  • A
    $512$
  • B
    $272$
  • ✓
    $384$
  • D
    $264$
✓

Answer

Correct option: C.
$384$
c
$\sum\limits_{n = 0}^4 {{{\left( {1009 - 2n} \right)}^4}} {\,^4}{C_n}{\left( { - 1} \right)^n}$

$(1009)^{4}-4(1007)^{2}+6 \cdot(1005)^{4}-4(1003)^{4}+(1001)^{4}$

$(1005+4)^{4}+(1005-4)^{4}$

$-4\left[(1005+2)^{4}+(1005-2)^{4}\right]+6(1005)^{4}$

$=512-4 \times 32=384$

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