B. In how many different ways can 15 identical cookies be distributed among 4 children if each child receives at least 2 cookies and no more than 5 cookies? Because we will distribute at least 2 but no more than 5 blocks among 4 children then the generating function is:(x2+x3+x4+x5) (x2(1+x+ x2+x3)) (x²((1-x)/(1-x))) x8 ((1-x)/(1-x))4We need the coefficient of x15 in this product. [x15] x8 ((1-x)/(1-x))^4We can reduce it to x7 [x7] (1-x4)4 (1-x)-4a7= C(4,3)C(6,3) + C(4,4) C(10,7) //


To solve the problem of distributing 15 identical cookies among 4 children with the condition that each child receives between 2 and 5 cookies, we use a combinatorial approach involving generating functions. Let's break down the process step by step, following the expression you provided.

### Step-by-step Breakdown:

#### 1. **Generating Function Setup:**
The children can receive 2, 3, 4, or 5 cookies. Therefore, the generating function for the number of ways each child can receive cookies is:

\[
(x^2 + x^3 + x^4 + x^5)
\]

#### 2. **Generating Function for All 4 Children:**
The total generating function for all 4 children is the product of the individual generating functions:

\[
(x^2 + x^3 + x^4 + x^5)^4
\]

#### 3. **Factoring the Generating Function:**
We can express the sum \(x^2 + x^3 + x^4 + x^5\) as:

\[
x^2(1 + x + x^2 + x^3)
\]

Thus, the generating function becomes:

\[
(x^2(1 + x + x^2 + x^3))^4
\]

This can be factored further into:

\[
x^8(1 + x + x^2 + x^3)^4
\]

#### 4. **Expanding the Inner Sum:**
The expression \(1 + x + x^2 + x^3\) can be rewritten as:

\[
\frac{1 - x^4}{1 - x}
\]

Thus, the generating function is now:

\[
x^8 \left( \frac{1 - x^4}{1 - x} \right)^4
\]

#### 5. **Finding the Coefficient of \(x^{15}\):**
We now need the coefficient of \(x^{15}\) in the expanded product. To do this, we can rewrite the expression as:

\[
x^8 \left( (1 - x^4)^4 (1 - x)^{-4} \right)
\]

This simplifies to:

\[
x^8 (1 - 4x^4 + \text{higher powers}) (1 - x)^{-4}
\]

We are tasked with finding the coefficient of \(x^7\) in \((1 - 4x^4)(1 - x)^{-4}\).

#### 6. **Expanding \((1 - x)^{-4}\):**
The binomial expansion of \((1 - x)^{-4}\) is given by:

\[
(1 - x)^{-4} = \sum_{n=0}^{\infty} \binom{n+3}{3} x^n
\]

Thus, the first few terms are:

\[
1 + 4x + 10x^2 + 20x^3 + 35x^4 + 56x^5 + 84x^6 + 120x^7 + \cdots
\]

#### 7. **Combining with \(1 - 4x^4\):**
We now multiply \((1 - 4x^4)\) by \((1 - x)^{-4}\):

\[
(1 - 4x^4)(1 + 4x + 10x^2 + 20x^3 + 35x^4 + 56x^5 + 84x^6 + 120x^7 + \cdots)
\]

This gives:

\[
1 + 4x + 10x^2 + 20x^3 + (35 - 4)x^4 + 56x^5 + 84x^6 + (120 - 4 \times 35)x^7 + \cdots
\]

Simplifying the terms relevant for \(x^7\):

\[
1 + 4x + 10x^2 + 20x^3 + 31x^4 + 56x^5 + 84x^6 + (120 - 140)x^7 = 1 + 4x + 10x^2 + 20x^3 + 31x^4 + 56x^5 + 84x^6 - 20x^7
\]

#### 8. **Finding the Coefficient of \(x^7\):**
The coefficient of \(x^7\) is \(-20\).

#### 9. **Conclusion:**
Therefore, the number of ways to distribute the 15 cookies under the given constraints is:

\[
\boxed{-20}
\]

Let me know if you'd like further details or a different explanation!

Here are 5 related questions:
1. What if we had to distribute 20 cookies instead of 15 under the same constraints?
2. How does the generating function method generalize to any number of children and constraints?
3. What is the combinatorial interpretation of the binomial expansion used here?
4. Can you derive the generating function for a different cookie distribution problem where each child receives between 1 and 3 cookies?
5. How would the approach change if there was no upper limit on the number of cookies each child could receive?

**Tip**: When using generating functions, always check if a sum can be simplified into a well-known series like a geometric series!

In how many different ways can 15 identical cookies be distributed among 4 children if each child receives at least 2 cookies and no more than 5 cookies?

Explore the solution to distributing 15 identical cookies among 4 children, where each child receives between 2 and 5 cookies. This problem involves advanced combinatorial techniques such as generating functions and binomial expansions, suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation