Among the inequalities below,which ones are t | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) $I. n! \leq n^n$ is true because $\frac{n^n}{n!} = \frac{n}{n} 	imes \frac{n}{n-1} 	imes \dots 	imes \frac{n}{1} \geq 1$.$II. (n!)^2 \leq n^n$

Vedclass · Accepted Answer

$I, III$ and $IV$

Vedclass · Answer

(B) $I. n! \leq n^n$ is true because $\frac{n^n}{n!} = \frac{n}{n} 	imes \frac{n}{n-1} 	imes \dots 	imes \frac{n}{1} \geq 1$.$II. (n!)^2 \leq n^n$ is false. For large $n$,$(n!)^2$ grows much faster than $n^n$.$III. 10^n \leq n!$ is true for $n > 1000$ because the product $\frac{n}{10} 	imes \frac{n-1}{10} 	imes \dots 	imes \frac{1}{10}$ will eventually exceed $1$ as $n$ increases.$IV. n^n \leq (2n)!$ is true. Since $(2n)! = 1 	imes 2 	imes \dots 	imes n 	imes (n+1) 	imes \dots 	imes 2n$,it is clearly much larger than $n^n = n 	imes n 	imes \dots 	imes n$ ($n$ times).Thus,$I, III,$ and $IV$ are true.

Among the inequalities below,which ones are true for all natural numbers $n > 1000$?
$I. n! \leq n^n$
$II. (n!)^2 \leq n^n$
$III. 10^n \leq n!$
$IV. n^n \leq (2n)!$

Similar Questions

Number of $4-$digit numbers (the repetition of digits is allowed) which are made using the digits $1, 2, 3$ and $5$,and are divisible by $15$,is equal to $............$.

If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero,then $n$ is equal to:

The sum of all the $4$-digit numbers formed by taking all the digits from ${2, 3, 5, 7}$ without repetition is:

The number of $3$-$digit$ odd numbers,whose sum of digits is a multiple of $7$,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Among the inequalities below,which ones are true for all natural numbers $n > 1000$?$I. n! \leq n^n$$II. (n!)^2 \leq n^n$$III. 10^n \leq n!$$IV. n^n \leq (2n)!$