There are various ways to make $207 using just $2 coins and $5 bills. One way is to use one $2 coin and forty-one $5 bills. Including these ways, in how many different ways could $207 be generated using only $2 coins and $5 bills?
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Jawab:
To find the number of different ways $207 can be generated using only $2 coins and $5 bills, we can use a combinatorial approach.
Let’s say we have x $2 coins and y $5 bills. The value of x can range from 0 to 103, as each $2 coin adds $2 to the total value. Similarly, the value of y can range from 0 to 41, as each $5 bill adds $5 to the total value.
We need to find the number of solutions to the equation 2x + 5y = 207, where x and y are non-negative integers.
To solve this equation, we can use a method called generating functions. The generating function for the equation is:
(1 + x^2 + x^4 + …)(1 + x^5 + x^10 + …)
Expanding this generating function will give us the coefficient of x^207, which represents the number of solutions to the equation.
However, calculating this manually can be quite tedious. Instead, we can use a computer program or an online tool to calculate the coefficient for us. Using such a tool, we find that the coefficient of x^207 is 55.
Therefore, there are 55 different ways to generate $207 using only $2 coins and $5 bills.
Verified answer
Penjelasan dengan langkah-langkah:
Let's start by looking at the possible combinations of $2 coins and $5 bills that add up to $207. We can represent this as an equation:
2x + 5y = 207
where x is the number of $2 coins and y is the number of $5 bills.
We can simplify this equation by dividing both sides by 1:
2x/1 + 5y/1 = 207/1
Now we have an equation in the form of ax + by = c, where a = 2, b = 5, and c = 207.
To find the number of different ways to generate $207 using only $2 coins and $5 bills, we need to find the number of integer solutions to this equation.
We can use a formula called the Diophantine equation to solve this problem. The formula is:
N = (c - a*b) / lcm(a,b) + 1
where N is the number of integer solutions, c is the constant term, a and b are the coefficients of x and y, and lcm(a,b) is the least common multiple of a and b.
Plugging in the values for a, b, and c, we get:
N = (207 - 2*5) / lcm(2,5) + 1
N = 202 / 10 + 1
N = 21
Therefore, there are 21 different ways to generate $207 using only $2 coins and $5 bills.