How many ways are there to make cent's using dimes and nickels?

To find the number of ways to make a certain amount of cents using dimes (10 cents) and nickels (5 cents), we can use the equation (10d + 5n = c), where (d) is the number of dimes, (n) is the number of nickels, and (c) is the total amount in cents. Rearranging gives (n = \frac{c - 10d}{5}). The number of non-negative integer solutions depends on the maximum number of dimes that can be used, which is (\lfloor c/10 \rfloor). For each value of (d), there is a corresponding value of (n) as long as (c - 10d) is non-negative and divisible by 5.