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STD 11 - 6. permutation and combination — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 6. permutation and combination4 Marks
MCQ
$\mathop \sum \limits_{0 \le i < j \le n} i\left( \begin{array}{l} n\\ j \end{array} \right)$ is equal to
  • A
    $n^22^{n-1}$
  • B
    $(n^2 -1)2^{n-1}$
  • C
    $(n-1)^22^n$
  • ✓
    $n(n-1)2^{n-3}$
✓

Answer

Correct option: D.
$n(n-1)2^{n-3}$
$\sum\limits_{0 \le i < j \le n} i \left( {\frac{n}{j}} \right) $
$= \frac{{n\left( {n - 1} \right)}}{2}\sum\limits_{k = 0}^n {\left( {\frac{{n - 2}}{k}} \right)} $
$=\frac{n(n-1)}{2} 2^{n-2}$
$=n(n-1) 2^{n-3}$

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