Question

Let 5 digit numbers be constructed using the digits $0,2,3,4,7,9$ with repetition allowed, and are arranged in ascending order with serial numbers Then the serial number of the number 42923 is

Hint:

When constructing numbers with restricted digits, organize them systematically to calculate the serial number efficiently. Use counting principles like combinations and arrange them in ascending order to avoid missing any possibilities.

Answer 1

Correct Answer: 2997

The problem asks us to find the serial number of the number 42923 when 5-digit numbers are constructed from the digits {0, 2, 3, 4, 7, 9} with repetition allowed, arranged in ascending order. Let's break down the steps to solve this.

First, note the total number of available digits: 6 (0, 2, 3, 4, 7, 9). Numbers must be 5 digits long. We won't allow numbers starting with 0, as they wouldn't be valid 5-digit numbers.

Step 1: Count numbers starting with digits less than 4
For the first digit being 2 or 3:

• First digit = 2 or 3: 2 choices (2, 3)
• Each of the remaining 4 digits can be any of the 6 digits: 6⁴ choices

Thus, the total numbers starting with 2 or 3 = 2 × 6⁴ = 2592.

Step 2: Count numbers starting with 42 and digits less than 9
Remaining digits for 42xx9: all positions have 6 choices, but last digit is less.

• Position 3: 0, 2, 3, 4, 7
• Position 4: any of the 6

Total for starting 42 = 5 × 6³ = 5 × 216 = 1080.

Step 3: Specify starting conditions for 42xxx
For digit 3 = 2, number will be in form 429xx.
Digit 3 = 9, number 42923 is in form 4292x:

• Last digit choices = 5 (0, 2, 3, 4, 7)
• Numbers form: 42920, 42922, 42923

Thus, the position of 42923 in this is immediate after 42922, confirming it's the 5th entry.

Thus, to find the overall serial number:

1. Count numbers less than form 42(4 places)
2. 1s in 42x92xx counted earlier
3. Combine: 2592 + 1025 + 2 = 2997

The serial number of 42923 is 2997, fitting perfectly in the expected range.