### Instruction: How many ways are there to divide a set of 10 distinguishable objects into 4 non-empty indistinguishable groups?### Response: To solve this problem, we can use the Stirling numbers of the second kind, which are denoted by S(n, k) and represent the number of ways to partition a set of n distinguishable objects into k non-empty indistinguishable groups. In this case, we want to find S(10, 4).

However, since the groups are indistinguishable, we need to account for the over-counting that occurs due to the different permutations of the groups. To do this, we can use the following formula:

Number of ways = S(n, k) / k!

In our case, n = 10 and k = 4. Using a table or calculator for Stirling numbers of the second kind, we find that S(10, 4) = 34105. Now, we need to divide this by 4! to account for the indistinguishability of the groups:

Number of ways = 34105 / (4!) = 34105 / 24 = 1421

So, there are 1421 ways to divide a set of 10 distinguishable objects into 4 non-empty indistinguishable groups.