Final answer:
The question involves combinatorial calculations for different scenarios of creating collections of coins with various constraints on quantities. The stars and bars technique is commonly used, and constraints are applied for scenarios with limited quantities of certain coins.
Step-by-step explanation:
The question deals with the combinatorial problem of counting the number of ways to choose collections of 30 coins, considering various types of coins and restrictions on their quantities. The four sub-questions address different scenarios:
- (a) When there are at least 30 coins of each type, the problem is equivalent to finding non-negative integer solutions to the equation x + y + z + w = 30, which corresponds to a combinatorial problem commonly solved using stars and bars technique.
- (b) With only 15 quarters and at least 30 of the remaining types, the collections would have to be chosen with the constraint that at most 15 quarters can be included.
- (c) With only 20 dimes and at least 30 of the other types, the collections must accommodate the constraint that at most 20 dimes can be selected.
- (d) When there are only 15 quarters and 20 dimes but at least 30 pennies and nickels, the collections need to account for both constraints together.
For each scenario, we would calculate the different combinations accordingly, accommodating the respective constraints