Key Concepts and Formulas

• Lexicographical Ordering (Descending): Ordering elements (in this case, numbers) based on comparing their digits from left to right, with larger digits preceding smaller digits.
• Counting with Repetition: If there are $n$ choices for each of $k$ positions, the total number of possibilities is $n^k$.
• Serial Number: The serial number of an element in an ordered list is its position in the list, starting from 1. For a descending list, it is calculated as (Number of elements strictly greater than the element) + 1.

Step-by-Step Solution

We are given the digits {1, 2, 3, 5, 7}, and we want to find the serial number of 35337 when all possible five-digit numbers formed using these digits with repetition are arranged in descending order.

Step 1: Numbers starting with a digit greater than 3 (the first digit of 35337)

• Why this step: Any five-digit number starting with a digit greater than 3 will be larger than 35337. The digits greater than 3 are 5 and 7.
• Case 1: First digit is 7.
  • Calculation: The first digit is fixed as 7. The remaining four digits can each be any of the 5 digits {1, 2, 3, 5, 7}.
  • Number of such numbers = $1 \times 5 \times 5 \times 5 \times 5 = 5^4 = 625$.
• Case 2: First digit is 5.
  • Calculation: The first digit is fixed as 5. The remaining four digits can each be any of the 5 digits {1, 2, 3, 5, 7}.
  • Number of such numbers = $1 \times 5 \times 5 \times 5 \times 5 = 5^4 = 625$.

Step 2: Numbers starting with 3, and whose second digit is greater than 5 (the second digit of 35337)

• Why this step: If the first digit is the same (3), we compare the second digit. Numbers with the first digit as 3 and the second digit greater than 5 will be larger than 35337. The only digit greater than 5 in our set is 7.
• Case 3: First two digits are 37.
  • Calculation: The first two digits are fixed as 3 and 7. The remaining three digits can each be any of the 5 digits {1, 2, 3, 5, 7}.
  • Number of such numbers = $1 \times 1 \times 5 \times 5 \times 5 = 5^3 = 125$.

Step 3: Numbers starting with 35, and whose third digit is greater than 3 (the third digit of 35337)

• Why this step: If the first two digits are the same (35), we compare the third digit. Numbers with the first two digits as 35 and the third digit greater than 3 will be larger than 35337. The digits greater than 3 in our set are 5 and 7.
• Case 4: First three digits are 357.
  • Calculation: The first three digits are fixed as 3, 5, and 7. The remaining two digits can each be any of the 5 digits {1, 2, 3, 5, 7}.
  • Number of such numbers = $1 \times 1 \times 1 \times 5 \times 5 = 5^2 = 25$.
• Case 5: First three digits are 355.
  • Calculation: The first three digits are fixed as 3, 5, and 5. The remaining two digits can each be any of the 5 digits {1, 2, 3, 5, 7}.
  • Number of such numbers = $1 \times 1 \times 1 \times 5 \times 5 = 5^2 = 25$.

Step 4: Numbers starting with 353, and whose fourth digit is greater than 3 (the fourth digit of 35337)

• Why this step: If the first three digits are the same (353), we compare the fourth digit. Numbers with the first three digits as 353 and the fourth digit greater than 3 will be larger than 35337. The digits greater than 3 in our set are 5 and 7.
• Case 6: First four digits are 3537.
  • Calculation: The first four digits are fixed as 3, 5, 3, and 7. The remaining digit can be any of the 5 digits {1, 2, 3, 5, 7}.
  • Number of such numbers = $1 \times 1 \times 1 \times 1 \times 5 = 5$.
• Case 7: First four digits are 3535.
  • Calculation: The first four digits are fixed as 3, 5, 3, and 5. The remaining digit can be any of the 5 digits {1, 2, 3, 5, 7}.
  • Number of such numbers = $1 \times 1 \times 1 \times 1 \times 5 = 5$.

Step 5: Numbers starting with 3533, and whose fifth digit is greater than 7 (the fifth digit of 35337)

• Why this step: If the first four digits are the same (3533), we compare the fifth digit. Numbers with the first four digits as 3533 and the fifth digit greater than 7 will be larger than 35337.
• Since there are no digits in the set {1, 2, 3, 5, 7} that are greater than 7, there are no such numbers.

Step 6: Calculate the total count and serial number

• The total number of five-digit numbers strictly greater than 35337 is the sum of the counts from all the cases above: Total numbers > 35337 = $625 + 625 + 125 + 25 + 25 + 5 + 5 = 1435$.
• The serial number of 35337 is the count of numbers strictly greater than it, plus 1: Serial number of 35337 = $1435 + 1 = 1436$.

Step 7: Re-Examine the Answer Choices Since the problem is marked as having "Correct Answer: 1", and the derivation yielded 1436, there is likely an error in the problem statement or the provided answer. However, based on the problem description, the correct answer is 1436. Let's re-examine the problem statement to be absolutely sure.

"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 ____________."

The problem statement is clear, and the derivation is correct. There is likely an error in the provided "Correct Answer".

Common Mistakes & Tips

• Incorrect Digit Ordering: Ensuring the correct descending order is crucial. In this case, 7 > 5 > 3 > 2 > 1.
• Forgetting to Add 1: Remember to add 1 to the count of strictly greater numbers to get the serial number.
• Double Counting: Be careful not to double-count numbers in different cases. The cases should be mutually exclusive.

Summary

To find the serial number of 35337, we count all the five-digit numbers that are strictly greater than 35337 when arranged in descending lexicographical order. We systematically compare the digits from left to right and count the possibilities for each case where a digit is greater. Summing these counts and adding 1 gives us the serial number. Based on the calculations, the serial number of 35337 is 1436. However, the provided correct answer is 1, which is likely incorrect.

Final Answer

The final answer is \boxed{1436}.