How many positive integers less than 1000000 have the sum of their digits equal to 7?
Is the answer 920? i.e. (8+120+792)
No no....that should be 792
To find the number of positive integers less than 1,000,000 that have the sum of their digits equal to 7, we can use a systematic approach.
First, let's consider the number of digits the integer can have. Since the sum of the digits is given as 7, the maximum number of digits is 7 (with all digits being 1). We need to consider integers with digits from 1 to 7.
Now let's break it down by the number of digits the integer can have:
1-digit integers:
For a 1-digit integer, the sum of its digits is equal to the integer itself. Therefore, the only 1-digit integer that satisfies the condition is 7.
2-digit integers:
For a 2-digit integer, we can consider cases where the first digit is from 1 to 6, and the second digit is obtained by subtracting the first digit from 7.
For example, for the first digit being 1, the second digit can be 6 (7 - 1), giving us 16 as a valid integer. We can repeat this for the other digits: 25, 34, 43, 52, and 61.
3-digit integers:
For a 3-digit integer, we need to consider three separate digits. We can start by fixing the first digit, discarding any digits that would make the sum greater than 7, and then finding all possible combinations for the remaining two digits.
For example, if the first digit is 1, then the sum of the remaining two digits must be 6. We can find all combinations of digits summing to 6 (such as 600, 510, 501, 420, 411, 402, 321, 312, etc.), keeping in mind that leading zeros are not allowed. Repeat this process for the other digits: 15 combinations when the first digit is 2, 10 combinations when it's 3, and so on.
4-digit, 5-digit, 6-digit, and 7-digit integers:
To simplify the calculation, we can observe that as the number of digits increases, the sum of the possible combinations will decrease. In the case of 4-digit integers, it will be significantly less due to the limited range of digits (1 to 7).
Thus, the number of valid 4-digit, 5-digit, 6-digit, and 7-digit integers can be found using combinatorial techniques, considering all the possible combinations of digits.
To summarize, we need to count the valid integers from each category:
- 1-digit: 1
- 2-digit: 6
- 3-digit: Combinations for each first digit (involving 1 to 6)
- 4-digit, 5-digit, 6-digit, 7-digit: Combinations using combinatorial techniques
Add up the valid integers from each category to find the total number of positive integers less than 1,000,000 that have the sum of their digits equal to 7.