Find the Generating function for the number of ways to select 10 candy bars from large supplies of six different kinds.using generative functions

To solve the problem using generating functions, we aim to find the generating function that represents the number of ways to select 10 candy bars from large supplies of six different kinds of candy. Let's break this down step-by-step:

### Step 1: Formulate the problem
We have six different kinds of candy bars, say \( C_1, C_2, C_3, C_4, C_5, C_6 \). We need to find the number of ways to select 10 candy bars in total, which can be modeled as solving the equation:
\[
x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 10
\]
where \( x_i \) (for \( i = 1, 2, \dots, 6 \)) represents the number of candy bars of kind \( C_i \) chosen, and each \( x_i \geq 0 \).

### Step 2: Generating functions for each type of candy
For each type of candy, the possible number of bars we can pick is non-negative. Therefore, the generating function for each type of candy is:
\[
1 + x + x^2 + x^3 + \cdots = \frac{1}{1 - x}
\]
This generating function represents selecting 0 or more candy bars from a given type.

### Step 3: Total generating function
Since we have six types of candy, and we can select independently from each type, the total generating function is the product of the generating functions for each candy:
\[
G(x) = \left( \frac{1}{1 - x} \right)^6
\]
This function generates the number of ways to select candy bars, where the exponent of \( x^n \) gives the number of ways to select \( n \) candy bars.

### Step 4: Extract the coefficient of \( x^{10} \)
Now we need to find the coefficient of \( x^{10} \) in the expansion of \( G(x) = \left( \frac{1}{1 - x} \right)^6 \). This is equivalent to finding the number of ways to select 10 candy bars from the 6 types.

We can expand the function using the binomial series:
\[
\left( \frac{1}{1 - x} \right)^6 = \sum_{n=0}^{\infty} \binom{n+5}{5} x^n
\]
Thus, the coefficient of \( x^{10} \) is \( \binom{10 + 5}{5} = \binom{15}{5} \).

### Step 5: Calculate the binomial coefficient
We now calculate the binomial coefficient:
\[
\binom{15}{5} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = 3003
\]

### Final Answer:
The number of ways to select 10 candy bars from large supplies of 6 different kinds is **3003**.

---

Would you like more details on the binomial expansion, or do you have any questions?

Here are some related questions:
1. How does the binomial theorem relate to generating functions?
2. What is the significance of finding coefficients in generating functions?
3. How would the generating function change if there were a limit on how many candy bars of each type could be selected?
4. What if there were more than 6 types of candy—how would that affect the solution?
5. Can generating functions be used to solve problems involving restrictions on the selection?

**Tip:** Generating functions are a powerful tool for solving combinatorial problems, especially those involving partitions or selections with repetition.

Find the Generating function for the number of ways to select 10 candy bars from large supplies of six different kinds using generating functions.

Learn how to find the generating function to calculate the number of ways to select 10 candy bars from large supplies of six different kinds. This solution covers combinatorics, generating functions, and the binomial theorem, resulting in 3003 ways. Suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation