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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
The largest power of $2$ that divides $\frac{200 !}{100 !}$ is
  • A
    $98$
  • B
    $99$
  • ✓
    $100$
  • D
    $101$
✓

Answer

Correct option: C.
$100$
c
(c)

Exponent of $2$ in $200 !$.

$=\left[\begin{array}{c}200 \\ 2\end{array}\right]+\left[\begin{array}{c}200 \\ 2^2\end{array}\right]+\left[\begin{array}{c}200 \\ 2^3\end{array}\right]+\left[\begin{array}{c}200 \\ 2^4\end{array}\right]+\left[\begin{array}{c}200 \\ 2^5\end{array}\right] +\left[\frac{200}{2^8}\right]+\left[\begin{array}{c}200 \\ 2^7\end{array}\right]+\left[\begin{array}{c}200 \\ 2^8\end{array}\right]$

$=100+50+25+12+6+3+1=197$

Exponent of $2$ in $100!$

$=\left[\frac{100}{2}\right]+\left[\frac{100}{2^2}\right]+\left[\begin{array}{c}100 \\ 2^3\end{array}\right]+\left[\begin{array}{c}100 \\ 2^4\end{array}\right]+\left[\begin{array}{c}100 \\ 2^5\end{array}\right]+\left[\begin{array}{c}100 \\ 2^6\end{array}\right]+\left[\begin{array}{c}100 \\ 2^7\end{array}\right]$

$\operatorname{In} \frac{200 !}{100 !}=\frac{2^{197}}{2^{97}}=2^{100}$

$\therefore$ The largest power of $2$ is $100$.

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