Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in des

Q: Why do we add 1 at the end? It feels like we've already counted everything.

Think of it like a queue. If there are 1435 people standing in front of you, what is your position? You are the 1436th person. We count the numbers *strictly greater* than our target, so the target itself is the next one in line.

Q: Why is it $5^n$ and not $5 \times 4 \times 3 \dots$?

That is the trap! If repetition were *not* allowed, you would use permutations ($P(n,r)$). But the problem explicitly states 'repetition is allowed'. This means every slot has the full set of 5 digits available to it. Always read the question carefully—it's the difference between a correct answer and a silly mistake.

Q: Does the descending order change the math or just the direction?

It changes the direction of your logic. In ascending order, you count numbers *smaller* than the target. In descending order, you count numbers *larger* than the target. The math remains the same, but your perspective must shift. Always visualize the list: 77777 is at the top, 11111 is at the bottom.

Question

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is ____.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

To find the rank of the number $35337$ in a list of five-digit numbers formed using the set $\{1, 2, 3, 5, 7\}$ arranged in descending order, we must count every number strictly greater than $35337$.

Since the list is in descending order, the rank is defined as:
$$	ext{Rank} = (	ext{Count of numbers greater than } 35337) + 1$$

### The Philosophy of the First Digit

We examine the first digit of our target, which is $3$. To find numbers larger than $35337$, we consider numbers starting with a digit greater than $3$. From our set $\{1, 2, 3, 5, 7\}$, the digits greater than $3$ are $7$ and $5$.

For numbers starting with $7$, the pattern is $7 \_ \_ \_ \_$. Since repetition is allowed, each of the $4$ remaining slots can be filled in $5$ ways:
$$5^4 = 625 	ext{ numbers}$$

Similarly, for numbers starting with $5$, the pattern is $5 \_ \_ \_ \_$, which also yields:
$$5^4 = 625 	ext{ numbers}$$

The running total of numbers greater than our target is $625 + 625 = 1250$.

### Drilling Down: The Second and Third Digits

Next, we fix the first digit as $3$ and look at the second digit of our target, which is $5$. To find numbers larger than $35337$ starting with $3$, the second digit must be greater than $5$. The only digit greater than $5$ is $7$.

For the pattern $37 \_ \_ \_$, there are $3$ slots remaining, yielding:
$$5^3 = 125 	ext{ numbers}$$

Our running total is now $1250 + 125 = 1375$. Now, we fix the first two digits as $35$ and look at the third digit, which is $3$. To be larger, the third digit must be $7$ or $5$.

For the patterns $357 \_ \_$ and $355 \_ \_$, there are $2$ slots remaining for each:
$$2 	imes 5^2 = 2 	imes 25 = 50 	ext{ numbers}$$

Our running total is now $1375 + 50 = 1425$.

### The Final Stretch

We fix the first three digits as $353$. The fourth digit of our target is $3$. To be larger, the fourth digit must be $7$ or $5$.

For the patterns $3537 \_$ and $3535 \_$, there is $1$ slot remaining for each:
$$2 	imes 5^1 = 10 	ext{ numbers}$$

Our running total is now $1425 + 10 = 1435$.

Finally, we fix the first four digits as $3533$. The next number in the descending sequence is $35337$, which is our target.

We have counted $1435$ numbers strictly greater than $35337$. Therefore, the rank of $35337$ is:
$$	ext{Rank} = 1435 + 1 = \mathbf{1436}$$

Animated Solution for Mathematics - Permutations and Combinations: Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is ____.

Enter Numerical Value:

Visualized Solution

Understanding the Problem

The Logic of Descending Rank

First Digit Possibilities

Case 1: Numbers Starting with $7$

Case 2: Numbers Starting with $5$

Second Digit Possibilities

Case 3: Numbers Starting with $37$

Third Digit Possibilities

Case 4: Numbers Starting with $357$ or $355$

Fourth Digit Possibilities

Case 5: Numbers Starting with $3537$ or $3535$

The Final Digit

Final Calculation

The Sigma Insight: Linear Permutations

Analyzing the Setup

The Philosophy of the First Digit

Drilling Down

The Final Stretch

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