Final answer:
The total number of ways to choose 10 objects from 31, with 10 identical and 21 distinct, is the sum of combinations of the distinct objects for each number of identical objects chosen, resulting in 2^21 ways.
Step-by-step explanation:
The question pertains to the number of ways of choosing 10 objects out of 31, where 10 are identical and the remaining 21 are distinct. Using combinatorial principles, when dealing with identical objects and distinct objects, the scenario can be perceived as a combination problem. If all the objects were distinct, the answer would be a simple combination of 31 objects taken 10 at a time. However, since 10 objects are identical, they don't add extra combinations when selected.
To choose 10 objects from the 31 available, we could either choose 0, 1, 2, ..., or 10 of the identical objects, which would require us to choose 10, 9, 8, ..., or 0 of the distinct objects correspondingly. For each of these 11 scenarios, the number of ways to choose the distinct objects is the combination of 21 objects taken k at a time (C(21, k)), where k ranges from 0 to 10.
Therefore, the total number of ways to choose the 10 objects is the sum of all these combinations: C(21, 0) + C(21, 1) + C(21, 2) + ... + C(21, 10). Using the properties of combinations, the sum equates to 221, as each term corresponds to the expansion of (1+1)21.