How many ways can five types of coins add up to one dollar? The answer is exactly 292 distinct groupings — and finding that number is a combinatorics problem. A brute-force program tests every possible combination of pennies, nickels, dimes, quarters, and half-dollars. It works, but runs very slowly. As the target amount grows, the number of possible combinations explodes: ten thousand dollars yields over 666 quadrillion arrangements. That makes brute force impractical. When you search for patterns in the results, you find that the five-coin problem connects to simpler four-coin and three-coin problems. A new program built on those patterns runs over 26,000 times faster — solving the ten-thousand-dollar problem in under an hour.