Understand the ordering

Five-digit numbers are formed using the digits $\{1,2,3,5,7\}$ with repetition allowed.

Since they are arranged in descending order, the largest number gets serial number $1$.

Example: $77777$ has serial number $1$.

We need the serial number of $35337$.

Idea

The serial number of $35337$ is: $1 + \text{(number of 5-digit numbers greater than }35337)$

Since each position can be filled with any of $5$ digits, we count lexicographically from left to right, with digit priority: $7 > 5 > 3 > 2 > 1$

Count numbers greater than $35337$

We examine digit by digit.

First digit: $3$

Any number starting with a digit greater than $3$ will be larger.

Greater choices are $7,5$ i.e. $2$ choices.

For each such choice, remaining $4$ places can be filled in $5^4$ ways.

$2 \cdot 5^4 = 2 \cdot 625 = 1250$

Second digit: $5$

Now fix first digit as $3$.

Second digit greater than $5$ is only $7$, i.e. $1$ choice.

Remaining $3$ places can be filled in $5^3$ ways.

$1 \cdot 5^3 = 125$

Running total: $1250+125=1375$

Third digit: $3$

Fix first two digits as $35$.

Third digit greater than $3$ are $7,5$, i.e. $2$ choices.

Remaining $2$ places can be filled in $5^2$ ways.

$2 \cdot 5^2 = 50$

Running total: $1375+50=1425$

Fourth digit: $3$

Fix first three digits as $353$.

Fourth digit greater than $3$ are $7,5$, i.e. $2$ choices.

Remaining $1$ place can be filled in $5$ ways.

$2 \cdot 5 = 10$

Running total: $1425+10=1435$

Fifth digit: $7$

Fix first four digits as $3533$.

Fifth digit greater than $7$: none.

So contribution is $0$.

Find serial number

Hence, the number of numbers greater than $35337$ is $1435$

Therefore its serial number is $1435+1 = 1436$

Final answer

$\boxed{1436}$