how many permutations are there for 4 numbers (0,2,3 and 5) adding up to 25 in 8 turns?
To find the number of permutations for 4 numbers (0, 2, 3, and 5) adding up to 25 in 8 turns, we can use a combination of techniques called generating functions and coefficient extraction.
Step 1: Define the Generating Function
We can define a generating function for the given problem, where the exponent of each term represents the number of times we choose a particular number. In this case, we have four numbers, so our generating function will be:
G(x) = (1 + x^0)(1 + x^2)(1 + x^3)(1 + x^5)
Step 2: Simplify the Generating Function
Next, we simplify the generating function by multiplying the terms:
G(x) = (1 + x^2)(1 + x^3 + x^5 + x^8)
Step 3: Extract the Coefficient
To find the coefficient of x^25, we can expand the simplified generating function and find the term with x^25:
G(x) = 1 + x^2 + x^3 + x^5 + x^8 + x^5 * x^2 + x^8 * x^2 + x^8 * x^3 + x^8 * x^5
By simplifying further, we get:
G(x) = 1 + x^2 + x^3 + 2x^5 + 2x^7 + 2x^10 + x^13 + x^15 + x^18
The coefficient of x^25 is 0 since there is no term with a power of 25.
Therefore, there are 0 permutations of the numbers 0, 2, 3, and 5 that add up to 25 in 8 turns.