How Many Numbers Between 99 and 1000 Have Exactly One Digit 7?
When analyzing the range of numbers between 99 and 1000, a key question arises: how many of these numbers contain the digit 7 exactly once? This article provides a detailed breakdown of the methods and calculations involved, accompanied by practical examples to enhance understanding.
Understanding the Range
The given range is effectively between 100 and 999, which are all 3-digit numbers. The challenge is to find those numbers that have exactly one digit as 7. This involves examining each place digit (hundreds, tens, and units) and determining how many valid combinations there are where 7 appears exactly once.
Calculating the Total Count
Let's break down the calculation step-by-step:
Form 1: Number is 7ab
In this form, the digit 7 is fixed in the hundreds place, and the tens and units are variable (0-9, excluding 7).
There are 9 possible values for 'a' (0-9 excluding 7) and 9 possible values for 'b' (0-9 excluding 7).
Total numbers in this form 9 * 9 81
Form 2: Number is a7b
Here, the digit 7 is in the tens place, and the hundreds and units are variable (0-9, excluding 7).
There are 8 possible values for 'a' (1-9 excluding 7) and 9 possible values for 'b' (0-9 excluding 7).
Total numbers in this form 8 * 9 72
Form 3: Number is ab7
Similarly, the digit 7 is in the units place, and the hundreds and tens are variable (0-9, excluding 7).
There are 8 possible values for 'a' (1-9 excluding 7) and 9 possible values for 'b' (0-9 excluding 7).
Total numbers in this form 8 * 9 72
Combining all three forms, the total number of 3-digit numbers with exactly one digit 7 is: 81 72 72 225.
The “Sloppy” Counting Method Revisited
The initial method was indeed “sloppy” in its counting approach. Here's a corrected breakdown:
Between 99 and 200
There are 19 numbers with at least one 7 in the following: 107, 117, 127, 137, 147, 157, 167, 170-179, 187, 197. (19)
Between 200 and 600, 800, and 900
For each of these ranges (200-299, 300-399, ..., 800-899, 900-999), there are 19 numbers with at least one 7. Multiplying by 5 ranges (200-600, 800-900) gives 19 * 5 95.
Between 700 and 799
Here, the digit 7 is already fixed in the hundreds place, providing 100 numbers (700-799).
Total numbers in this range 100.
Final Calculation
Summing all the valid counts: 19 95 100 214.
Correct Total Count
The corrected total count is 214, acknowledging the presence of 7 in the fixed part and the variable digits.
Verification Using a Brute Force Approach
In J programming language, the calculation is straightforward and confirms the result of 225. The brute force approach confirms that there are 225 such numbers between 99 and 1000 where the digit 7 appears exactly once.
Conclusion
The refined, accurate method for determining the number of integers between 99 and 1000 that include the digit 7 exactly once is 225. This precise calculation accounts for the changes and ensures there are no overlaps or miscounts.