Please help me with the following problems...
1) How many ways can you make a change for a quater using an unlimited source of nickels, dimes, and pennies?
2) How many doubled digit numbers can you make with a least two identical numbers from 2-200. Not including 2 and 200. For example: 22 and 122
Sure, I can help you with these problems. Let's break them down and explain how to solve each one.
1) To find the number of ways to make change for a quarter using nickels, dimes, and pennies, we can use a technique called dynamic programming. We start by defining a list or array to keep track of the number of ways to make each value from 0 to 25 cents. We initialize the first element of the list to 1, because there is only one way to make 0 cents.
Then, we iterate through the denominations (nickels, dimes, and pennies) and for each denomination, we update the list in two steps. First, we check if the current denomination is less than or equal to the value we are considering. If it is, we add the number of ways to make the remaining value (value minus the denomination) using the current denomination. Second, we update the number of ways to make the current value by adding the number of ways we just calculated.
Finally, the last element in the list will give us the total number of ways to make a quarter using nickels, dimes, and pennies.
2) To find the number of doubled-digit numbers with at least two identical numbers from 2 to 200 (excluding 2 and 200), we can use the concept of combinations.
We can start by subtracting 1 from 200 to account for the exclusion of the numbers 2 and 200. This gives us a range from 1 to 199.
Next, we can calculate the number of combinations with two identical numbers, which can be done by choosing any two numbers from the range from 1 to 9 (including duplicates). This can be done using the formula for combinations: nCr = n! / (r! * (n-r)!)
Since we want to choose 2 numbers from 9, we can calculate the number of combinations as 9! / (2! * (9-2)!).
Finally, we add this result to the number of combinations with three identical numbers, which can be found by choosing any three numbers from the range from 1 to 9 (including duplicates). This can be calculated using the formula mentioned earlier with n = 9 and r = 3.
To find the total number of doubled-digit numbers, we add the number of combinations with two identical numbers and the number of combinations with three identical numbers.
I hope this explanation helps! Let me know if there's anything else I can assist you with.