© All Rights reserved @ LearnWithDash
Step-by-step Solution
Step 1: Understand the Problem and the Digits Involved
We need to form five-digit numbers using the digits 1, 2, 3, 5, 7 with repetitions allowed. These numbers are arranged in descending order, and the goal is to find the position (serial number) of the specific number 35337 in that ordered list.
Step 2: Recognize the Descending Order of Digits
The digits in descending order are 7, 5, 3, 2, 1. Hence, the largest possible five-digit number from these digits is 77777, and that is assigned serial number 1.
Step 3: Count All Numbers Starting With 7
All five-digit numbers beginning with 7 can use any of the 5 digits (1, 2, 3, 5, 7) in each of the remaining four positions. Thus, there are 5^4 numbers starting with 7.
Step 4: Count All Numbers Starting With 5
After exhausting numbers starting with 7, we move to numbers beginning with 5. Similarly, each of the remaining four positions can be any of the 5 digits. That also makes 5^4 possible numbers starting with 5.
Step 5: Count All Numbers Starting With 3 and Then Having Second Digit 7
Now we look at numbers starting with 3, in which the second digit is 7 (i.e., numbers of the form 37___). The remaining three positions can each be any of the 5 digits. Therefore, there are 5^3 numbers starting with 37.
Step 6: Count All Numbers Starting With 3, Then 5, and Then 7 (i.e., 357__)
Next, consider numbers beginning with 357. The last two positions can each be any of the 5 digits, giving 5^2 possibilities.
Step 7: Count All Numbers Starting With 3, Then 5, and Then 5 (i.e., 355__)
Numbers of the form 355__ also have 5^2 possibilities, as the remaining two positions can each be any of the 5 digits.
Step 8: Count All Numbers Starting With 3, 5, 3, 7 (i.e., 3537_)
For numbers of the form 3537_, the last digit can be any of the 5 digits. Hence, 5^1 = 5 possibilities.
Step 9: Count All Numbers Starting With 3, 5, 3, 5 (i.e., 3535_)
For numbers of the form 3535_, the last digit can also be any of the 5 digits, giving again 5 possibilities.
Step 10: Identify the Number 35337
After listing those 5 possibilities, the next number in descending order will be 35337. This is precisely the number whose position we seek. Hence, we add 1 more to account for its position right after those 5 possibilities.
Step 11: Summation to Get the Final Position
The total count of all numbers that come before 35337 plus 1 for the number 35337 itself is:
5^4 + 5^4 + 5^3 + 5^2 + 5^2 + 5 + 5 + 1
= 625 + 625 + 125 + 25 + 25 + 5 + 5 + 1
= 1436
Step 12: Conclude the Result
The serial number of the number 35337 in the descending-ordered list of all five-digit numbers formed by 1, 2, 3, 5, and 7 is 1436.
