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 this problem using generating functions, we need to find the generating function for the number of ways to select 10 candy bars from large supplies of six different kinds. This is a typical example of a problem involving **unrestricted selections**, which can be modeled using generating functions.

### Step 1: Understand the Problem
You are given six different types of candy bars (let's call them \( A_1, A_2, A_3, A_4, A_5, A_6 \)), and you want to select 10 candy bars from these. Each type of candy is available in an unlimited supply, meaning that there are no restrictions on how many bars of each type can be chosen.

This is a classic **stars and bars** problem, where the goal is to find how to distribute 10 indistinguishable candy bars among 6 distinguishable types.

### Step 2: Construct the Generating Function
For each type of candy, the number of candy bars you can select is an integer ranging from 0 to infinity (since there’s an unlimited supply). The generating function for selecting candy bars of type \( A_i \) is the following power series:

\[
1 + x + x^2 + x^3 + \cdots = \frac{1}{1 - x}
\]

This is the generating function for the number of ways to choose any non-negative number of candy bars of one kind.

Since you have six different kinds of candy bars, and the selection from each kind is independent, the generating function for the total selection is the product of six generating functions:

\[
\text{Total generating function} = \left(\frac{1}{1 - x}\right)^6 = \frac{1}{(1 - x)^6}
\]

### Step 3: Extract the Coefficient of \( x^{10} \)
The generating function \( \frac{1}{(1 - x)^6} \) represents the number of ways to select \( n \) candy bars from six kinds for any non-negative integer \( n \). The problem specifically asks for the number of ways to select 10 candy bars, so we need to find the coefficient of \( x^{10} \) in this expansion.

This corresponds to the binomial coefficient in the expansion of \( (1 - x)^{-6} \), which is given by the generalized binomial theorem:

\[
(1 - x)^{-6} = \sum_{n=0}^{\infty} \binom{n+5}{5} x^n
\]

Thus, the coefficient of \( x^{10} \) is:

\[
\binom{10 + 5}{5} = \binom{15}{5}
\]

### Step 4: Calculate the Binomial Coefficient
Now, we calculate \( \binom{15}{5} \):

\[
\binom{15}{5} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = 3003
\]

### Final Answer:
The generating function for the problem is \( \frac{1}{(1 - x)^6} \), and the number of ways to select 10 candy bars from six different kinds is \( 3003 \).

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Would you like further explanation on the generating function concept, or details on calculating binomial coefficients? Here are five related questions for you:

1. What is the generating function for selecting \( n \) candy bars from 4 kinds instead of 6?
2. How does the stars and bars method relate to generating functions?
3. How can generating functions be used to solve problems with restricted selections?
4. Can generating functions be applied to problems involving partitions of integers?
5. How do generating functions differ from recurrence relations in counting problems?

**Tip:** In combinatorics, generating functions are a powerful tool for solving problems involving unrestricted selections, especially when paired with binomial coefficients.

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

Discover how to use generating functions to find the number of ways to select 10 candy bars from 6 different types. This problem involves key concepts such as combinatorics, stars and bars, and the binomial coefficient theorem. A detailed step-by-step explanation for college-level students.

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