To find the number of five-digit positive integers where the digits are non-increasing from left to right, we need to count the possible combinations of digits.
The digits in such a number must satisfy the condition that each subsequent digit is less than or equal to the previous one. For example, in the number 78987, the digits decrease from left to right (7 ≥ 8, 8 ≥ 9, 9 ≥ 8).
To calculate the number of valid combinations, we can use a technique known as "stars and bars" or "balls and urns." Imagine we have five positions for digits, represented by stars, and four dividers (bars) to separate them into five groups.
For example, if the number is 55555, it would be represented as: **|**|**|**|**|**
Each group between the dividers represents a digit in the number, and the number of stars in each group represents the value of that digit.
Now, we need to count how many ways we can arrange these stars and bars. The number of ways is equivalent to the number of combinations of five stars and four bars, which can be calculated using the binomial coefficient formula:
Number of combinations = (n+k / k)
where n is the total number of items (stars + bars) and k is the number of items we want to choose (in this case, k = 4 bars).
So, there are 126 five-digit positive integers where the digits are non-increasing from left to right.