Given numbers are $1, 2, 3, 5, 7$.

And also they are arranged in descending order.

i.e., the numbers with four and five digits are included.

The number of five-digit number$=5!$

The number of four digits number

$={}^{5}p_4=\frac{5!}{(5-4)!}=\frac{120}{1}=120$

The number of $3$ digits starting with $7=$

i.e., $4$ digits have to be fit in $2$ place in ${}^{4}p_2$ ways,

${}^{4}p_2=\frac{4!}{(4-2)!}=4\times 3=12$

The number of $3$ digits numbers starting with $5=$

${}^{4}p_2=\frac{4!}{(4-2)!}=4\times 3=12$

The number of digits starting with $\underline{3}$

i.e., $3$ numbers have to fit in $1$ place in ${}^{3}p_1$ way.

${}^{3}p_1=\frac{3!}{(3-1)!}=\frac{3\times 2\times 1}{2\times 1}=3=3$

The number of digits starting with $\underset{¯}{3}\underset{¯}{5}$ is,

${}^{3}p_1=\frac{3!}{(3-1)!}=\frac{3\times 2\times 1}{2\times 1}=3$

And the next number is $327$.

$\therefore$ The rank of $327$ is

$=120+120+12+12+3+3+1$

$=271$

$\therefore$ Rank of $327=271$