Six problems (numbered 1 to 6) are set for a math test. A student can score 0, 1, 2, 3 points for each problem. Find the number of ways to score a total of 15 points for the six problems.
To solve this problem, we can use the concept of generating functions.
First, let's represent the possible scores for each problem as a polynomial. Since the student can score 0, 1, 2, or 3 points for each problem, the polynomial would be:
P(x) = 1 + x + x^2 + x^3
This polynomial represents the possible scores for a single problem.
Now, let's find the generating function G(x) for the total score of all six problems. We can do this by multiplying the polynomial P(x) six times, since there are six problems:
G(x) = P(x) * P(x) * P(x) * P(x) * P(x) * P(x)
= [1 + x + x^2 + x^3] * [1 + x + x^2 + x^3] * [1 + x + x^2 + x^3] * [1 + x + x^2 + x^3] * [1 + x + x^2 + x^3] * [1 + x + x^2 + x^3]
Next, we need to find the coefficient of the term x^15 in the expansion of G(x). This will give us the number of ways to score a total of 15 points.
Expanding G(x) is a lengthy process, but we can use algebraic software such as Mathematica or Wolfram Alpha to find the answer. Alternatively, we can use the method of partial fractions to simplify G(x) before expanding.
After expanding G(x), we can look at the term with the coefficient of x^15 and find its coefficient. This coefficient will give us the number of ways to score a total of 15 points for the six problems.
To find the number of ways to score a total of 15 points for the six problems, we can use the concept of generating functions. We will represent each problem as a variable in a polynomial and find the coefficient of the term with a total exponent of 15.
Step 1: Set up the generating function
Let's define the generating function for each problem as follows:
P(x) = 1 + x + x^2 + x^3
This represents the possible scores for each problem, where the exponent represents the number of points scored (0, 1, 2, or 3). We are interested in the coefficient of the term with a total exponent of 15.
Step 2: Multiply the generating functions
Since we need to find the number of ways to score a total of 15 points for the six problems, we multiply the generating functions for each problem:
P(x) * P(x) * P(x) * P(x) * P(x) * P(x)
Step 3: Expand the polynomial
Multiplying the generating functions will give us a polynomial. By expanding this polynomial, we can find the term with a total exponent of 15.
(P(x))^6
= (1 + x + x^2 + x^3)^6
To expand this polynomial efficiently, we can use the binomial theorem, which states that for any positive integer n:
(a + b)^n = C(n,0) * a^n * b^0 + C(n,1) * a^(n-1) * b^1 + ... + C(n,n-1) * a^1 * b^(n-1) + C(n,n) * a^0 * b^n
Where C(n,k) represents the binomial coefficient, also known as "n choose k".
So, applying the binomial theorem to our generating function:
(1 + x + x^2 + x^3)^6
= C(6,0) * 1^6 * (x^0)^6 + C(6,1) * 1^5 * (x)^1 + C(6,2) * 1^4 * (x^2)^2 + C(6,3) * 1^3 * (x^3)^3 + C(6,4) * 1^2 * (x^4)^4 + C(6,5) * 1^1 * (x^5)^5 + C(6,6) * 1^0 * (x^6)^6
= 1 + 6x + 21x^2 + 56x^3 + 126x^4 + 252x^5 + 462x^6 + 792x^7 + 1287x^8 + 2002x^9 + 3003x^10 + 4368x^11 + 6188x^12 + 8568x^13 + 11628x^14 + 15504x^15 + ...
From the expanded polynomial, we can see that the coefficient of the term with a total exponent of 15 is 15504.
Therefore, there are 15,504 ways to score a total of 15 points for the six problems.