How many different sums of money can U make with 4 pennies, two nickels and six quarters
no idea. Start listing them.
1 2 3 4
5 6 7 8 9
10 11 12 13 14 15
...
Seems like kind of a dumb question.
To find out how many different sums of money you can make, you can use a method called generating functions.
First, let's assign a value to each coin:
- A penny is worth 1 cent
- A nickel is worth 5 cents
- A quarter is worth 25 cents
Now, we can represent the number of each type of coin as variables in a generating function:
The generating function for pennies:
p(x) = (1 + x + x^2 + x^3 + x^4)
The generating function for nickels:
n(x) = (1 + x^5 + x^10)
The generating function for quarters:
q(x) = (1 + x^25 + x^50 + x^75 + x^100 + x^125 + x^150)
To find the generating function for the total value of coins, we multiply the generating functions together:
f(x) = p(x) * n(x) * q(x)
Expanding the terms of f(x) will give us the coefficients of each term, representing the number of ways to make different sums of money.
To find the answer, we need to find the coefficient of x^value, where value represents each possible sum of money. In this case, we are interested in the coefficients of the terms with 4 pennies, 2 nickels, and 6 quarters.
After multiplying and simplifying the generating functions, we get:
f(x) = x^4 * (1 + x + x^2 + x^3 + x^4) * (1 + x^5 + x^10) * (1 + x^25 + x^50 + x^75 + x^100 + x^125 + x^150)
Now, we need to find the coefficient of x^value in this expanded form:
The coefficient of x^value represents the number of ways to make the sum of value using the given coins.
Since manually expanding and finding the coefficient can be tedious, we can use math software like Mathematica or Python's SymPy library to simplify this process.
By plugging this generating function into a computer algebra system, we can find the coefficient of x^value, where value represents the different sums of money possible using 4 pennies, 2 nickels, and 6 quarters.