A vending machine had to be programmed to accept any combination of nickels, dimes, and quarters that totals 40 cents. How many different combinations are possible?
just list them:
q+d+n=40
q+3n=40
...
To find the number of different combinations possible, we can use a systematic approach.
Let's break down the problem into smaller subproblems:
1. Start with the largest coin, which is a quarter (25 cents).
2. Determine the number of quarters we can use, keeping in mind that the combination should total 40 cents.
Since the combination must have a total of 40 cents and quarters are 25 cents each, we can use between 0 and 1 quarters.
Now, let's move on to the smaller coins:
3. Consider the number of dimes (10 cents) we can use in the remaining amount. With 0 quarters, we have 40 cents remaining to consider.
- We can have anywhere between 0 and 4 dimes since each dime is worth 10 cents.
Finally, let's consider the smallest coin:
4. Determine the number of nickels (5 cents) we can use in the remaining amount. With 0 quarters and 0 dimes, we have 40 cents remaining.
- We can use anywhere between 0 and 8 nickels since each nickel is worth 5 cents.
Now, let's calculate the total number of combinations:
If we use 0 quarters, we have 5 possible numbers of dimes (0, 1, 2, 3, 4), and for each number of dimes, we have 9 possible numbers of nickels (0, 1, 2, 3, 4, 5, 6, 7, 8). This means we have 5 * 9 = 45 combinations.
If we use 1 quarter, we have 4 possible numbers of dimes (0, 1, 2, 3), and for each number of dimes, we have 5 possible numbers of nickels (0, 1, 2, 3, 4). This means we have 4 * 5 = 20 combinations.
Therefore, the total number of different combinations possible is 45 + 20 = 65.