Find the coefficient x
5
and x
10 for the expansion (1 − x)
4
(1 + x)
7
in ascending power of x. (Ans:

14, 3).

To find the coefficients of \(x^5\) and \(x^{10}\) in the expansion of \((1 - x)^4 (1 + x)^7\), we need to expand both binomials and then multiply the relevant terms to find the required coefficients.

### Step 1: Expand each binomial using the Binomial Theorem

1. **Expansion of \((1 - x)^4\):**
   \[
   (1 - x)^4 = \sum_{k=0}^{4} \binom{4}{k}(-x)^k = 1 - 4x + 6x^2 - 4x^3 + x^4
   \]

2. **Expansion of \((1 + x)^7\):**
   \[
   (1 + x)^7 = \sum_{k=0}^{7} \binom{7}{k}x^k = 1 + 7x + 21x^2 + 35x^3 + 35x^4 + 21x^5 + 7x^6 + x^7
   \]

### Step 2: Multiply the expansions and collect like terms

We multiply the expansions \((1 - x)^4\) and \((1 + x)^7\) and collect the coefficients of \(x^5\) and \(x^{10}\).

\[
(1 - 4x + 6x^2 - 4x^3 + x^4) \cdot (1 + 7x + 21x^2 + 35x^3 + 35x^4 + 21x^5 + 7x^6 + x^7)
\]

#### Coefficient of \(x^5\):
To find the coefficient of \(x^5\), consider the following products:
- \(1 \times 21x^5 = 21\)
- \(-4x \times 35x^4 = -140\)
- \(6x^2 \times 35x^3 = 210\)
- \(-4x^3 \times 21x^2 = -84\)
- \(x^4 \times 7x = 7\)

Adding these:
\[
21 - 140 + 210 - 84 + 7 = 14
\]
So, the coefficient of \(x^5\) is \(14\).

#### Coefficient of \(x^{10}\):
To find the coefficient of \(x^{10}\), consider the following products:
- \(x^4 \times x^6 = 1 \times 7 = 7 \)
- \(x^3 \times x^7 = -4 \times 1 = -4\)

Adding these:
\[
7 - 4 = 3
\]
So, the coefficient of \(x^{10}\) is \(3\).

### Final Answer:
The coefficient of \(x^5\) is **14**, and the coefficient of \(x^{10}\) is **3**.

Would you like a more detailed explanation, or do you have any other questions?

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Here are 5 related questions you might find interesting:
1. What is the general approach to finding coefficients in binomial expansions?
2. How does the Binomial Theorem apply to expansions of higher powers?
3. What is the significance of negative and positive coefficients in binomial expansions?
4. How do you find the coefficient of a specific term in the expansion of a trinomial?
5. What happens to the coefficients when you expand \((1-x)^n(1+x)^m\) for different values of \(n\) and \(m\)?

**Tip:** When dealing with binomial expansions, always look for patterns in coefficients, as they often follow Pascal's Triangle.

Learn how to find the coefficients of x^5 and x^10 in the expansion of (1 - x)^4 (1 + x)^7 with a detailed step-by-step solution. This problem covers the application of the Binomial Theorem, multiplication of binomials, and collection of like terms. Suitable for advanced high school students and those interested in algebraic expansions.

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