### Instruction: A math student needs to find the number of ways to divide a set of 7 objects into 4 non-empty groups. How many ways can this be done using Stirling numbers of the second kind?### Response: To solve this problem, we can use Stirling numbers of the second kind, denoted as S(n, k), which represent the number of ways to partition a set of n objects into k non-empty groups. In this case, we have n = 7 objects and k = 4 groups.

The Stirling number of the second kind can be found using the following formula:

S(n, k) = k * S(n-1, k) + S(n-1, k-1)

We can use this formula to find S(7, 4) by first calculating the necessary Stirling numbers for smaller values of n and k:

S(1, 1) = 1
S(2, 1) = 1
S(2, 2) = 1
S(3, 1) = 1
S(3, 2) = 3
S(3, 3) = 1
S(4, 1) = 1
S(4, 2) = 7
S(4, 3) = 6
S(4, 4) = 1
S(5, 1) = 1
S(5, 2) = 15
S(5, 3) = 25
S(5, 4) = 10
S(5, 5) = 1
S(6, 1) = 1
S(6, 2) = 31
S(6, 3) = 90
S(6, 4) = 65
S(6, 5) = 15
S(6, 6) = 1

Now we can find S(7, 4) using the formula:

S(7, 4) = 4 * S(6, 4) + S(6, 3)
S(7, 4) = 4 * 65 + 90
S(7, 4) = 260 + 90
S(7, 4) = 350

So there are 350 ways to divide a set of 7 objects into 4 non-empty groups using Stirling numbers of the second kind.