Math Problem Statement
Solution
The image shows a question related to the Pascal's Triangle (Pascalsches Dreieck in German). The problem asks to fully expand the expression using the binomial theorem with the help of Pascal's Triangle.
Binomial Expansion Formula:
The binomial expansion of is given by: where are binomial coefficients, which can be found using Pascal's Triangle.
Step-by-Step Expansion of :
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Identify the values:
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Pascal's Triangle for : The row for in Pascal's Triangle is: These are the binomial coefficients.
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Apply the Binomial Theorem: Using the binomial expansion formula, the expression becomes:
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Simplify each term:
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Final expansion:
So, the simplified form of is:
Do you need further details or have any questions?
Here are five related questions for further practice:
- What is the expansion of using Pascal's Triangle?
- How does Pascal's Triangle help in binomial expansions?
- What is the general formula for the binomial expansion of ?
- Can you explain the role of binomial coefficients in the expansion?
- How would you expand using Pascal's Triangle?
Tip: Pascal's Triangle can also be used to solve combinatorics problems, such as counting paths in a grid or selecting groups.
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Math Problem Analysis
Mathematical Concepts
Binomial Expansion
Algebra
Pascal's Triangle
Formulas
(a + b)^n = Σ [C(n, k) * a^(n-k) * b^k]
Pascal's Triangle coefficients
Theorems
Binomial Theorem
Suitable Grade Level
Grades 9-12