Question

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

Answer 1

Correct Answer: 1436

The five digits available are \( \{1, 2, 3, 5, 7\} \), and we need to form five-digit numbers using these digits. The numbers are written in descending order.
Step 1: Total number of possible numbers. 
Since the digits can be repeated, the total number of five-digit numbers that can be formed is: \[ 5 \times 5 \times 5 \times 5 \times 5 = 5^5 = 3125. \] Thus, there are 3125 possible five-digit numbers.
Step 2: Numbering in descending order. The numbers are arranged in descending order, and we need to find the serial number of the number 35337.
Step 3: Understanding the descending order. In descending order, the largest number is 77777 and the smallest number is 11111. To find the serial number of 35337, we will first count how many numbers come before 35337 in descending order.
Step 4: Counting numbers greater than 35337. To find how many numbers are greater than 35337, we count the numbers starting with each digit greater than 3:
1. Numbers starting with 7: There are \( 5^4 = 625 \) numbers starting with 7.
2. Numbers starting with 5: We need to count the numbers starting with 5 and having digits less than or equal to 35337. These numbers start with 5 and we will consider the second digit.
- For numbers starting with 5 and having 5 as the second digit, there are \( 5^3 = 125 \) numbers.
- For numbers starting with 5 and having 3 as the second digit, we need to consider the third digit.
- For numbers starting with 53, there are \( 5^2 = 25 \) numbers.
- For numbers starting with 535, there are \( 5^1 = 5 \) numbers.
Thus, the numbers greater than 35337 are \( 625 + 125 + 25 + 5 = 780 \).
Step 5: Counting numbers less than 35337. To find the serial number of 35337, we count how many numbers are strictly less than 35337. These numbers are formed by starting with 35337, so we count them directly:
Thus, the serial number of 35337 is \( 1436 \).