First slide

Permutations

Question

Let $E = [\frac{1}{3} + \frac{1}{50}] + [\frac{1}{3} + \frac{2}{50}] + [\frac{1}{3} + \frac{3}{50}] + \dots +$ up to 50 terms, then the exponent of 2 in E! is

Moderate

A

17
B

25
C

15
D

None of these

Solution

Let, $E = [\frac{1}{3} + \frac{x}{50}]$ where x = 1,2, ... ,50
For, $1 \leq x \leq 33, \frac{1}{3} < \frac{1}{3} + \frac{x}{50} < 1$
$∴ [\frac{1}{3} + \frac{x}{50}] = 0, for 1 \leq x \leq 33$
For $34 \leq x \leq 50, 1 < \frac{1}{3} + \frac{x}{50} < \frac{4}{3}$
$\Rightarrow [\frac{1}{3} + \frac{x}{50}] = 1$ , For $34 \leq x \leq 50$
Thus, E=17 and p=2
$\begin{array}{lc} ∵ & 2^{4} < 17 < 2^{5} \\ ∴ & a = 4 \end{array}$
Exponent of 2 in $(17)! = [\frac{17}{2}] + [\frac{17}{4}] + [\frac{17}{8}] + [\frac{17}{16}]$
$= 8 + 4 + 2 + 1 = 15$

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