Final answer:
To find the number of four-digit integers with a digit sum of 31, we can use the concept of stars and bars. Using the stars and bars formula, the number of ways to arrange the stars and bars is given by C(31+3, 3) = C(34, 3). Calculating this value, we find that there are 5,864 different four-digit integers with a digit sum of 31.
Step-by-step explanation:
To find the number of four-digit integers with a digit sum of 31, we can use the concept of stars and bars. Let's imagine having 31 stars representing the sum of the digits. We need to place 3 bars to separate the digits. Since there are 3 bars, we have 4 compartments to place the stars in. Each compartment represents a digit of the four-digit integer. Using the stars and bars formula, the number of ways to arrange the stars and bars is given by C(31+3, 3) = C(34, 3).
Using the formula for combinations, C(n, r) = n! / (r!(n-r)!), where n is the total number of objects and r is the number of objects chosen, we can find the number of combinations. Plugging in the values, we get C(34, 3) = 34! / (3!(34-3)!) = 34! / (3!31!).
Calculating this value, we find that there are 5,864 different four-digit integers with a digit sum of 31.