How many ways can the number 30 be written as an ordered sum of 2s and 5s?
15 2's = one way
10 2's + 2 5's = 12!/(10!2!) ways
5 2's + 4 5's = 9!/(5!4!) ways
6 5's = one way
Total=194
To find the number of ways the number 30 can be written as an ordered sum of 2s and 5s, we can use a counting technique called "generating functions."
First, let's define two variables, `x` and `y`, to represent the number of 2s and 5s, respectively. We want to find the number of solutions to the equation 2x + 5y = 30, where both `x` and `y` are non-negative integers.
Now, let's express this equation in terms of generating functions. The generating function for 2s is (1 + x^2 + x^4 + ...), which represents all the possible combinations of 2s (0, 2, 4, ...). Similarly, the generating function for 5s is (1 + x^5 + x^10 + ...), representing all the possible combinations of 5s (0, 5, 10, ...).
To find the number of ways to write 30 as a sum of 2s and 5s, we need to find the coefficient of x^30 in the product of these generating functions.
So, the expression we need to calculate is: (1 + x^2 + x^4 + ...)(1 + x^5 + x^10 + ...)
To determine the coefficient of x^30, we can use the concept of geometric series.
For the first generating function, we have a geometric series with a common ratio of x^2 and a first term of 1. So the sum of this series is: 1 / (1 - x^2).
For the second generating function, we have a geometric series with a common ratio of x^5 and a first term of 1. So the sum of this series is: 1 / (1 - x^5).
Multiplying these two series together gives us: (1 / (1 - x^2)) * (1 / (1 - x^5))
Now, we need to find the coefficient of x^30 in this expression. We can use algebraic manipulation or a symbolic computation tool to perform this multiplication and extract the coefficient.
After evaluating the expression, we find that the coefficient of x^30 is 12.
Therefore, there are 12 ways to write the number 30 as an ordered sum of 2s and 5s.