Final answer:
To ensure having at least three coins of each denomination from a change bowl containing ten quarters, twelve dimes, and fifteen nickels, one would need to grab 25 coins.
Step-by-step explanation:
The question asks for the minimum number of coins you must grab from a change bowl containing ten quarters, twelve dimes, and fifteen nickels to ensure that you have at least three of each kind of coin.
To guarantee having three of each, consider the worst-case scenario, where you take out all coins of one type, and then all of another. You could end up picking ten quarters and twelve dimes before you start picking nickels. Only after having these two stacks of coins would picking up nickels ensure you have three of them.
Therefore, the number of coins you need to guarantee three of each denomination is the full count of the two largest stacks of coins, plus three (for the three nickels you need). This gives you 10 (quarters) + 12 (dimes) + 3 (nickels) = 25 coins. So, if you grab 25 coins, you are certain to have at least three of each denomination.