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Expanding (x + y + z)^5
The expansion of (x + y + z)^5 is a challenging but rewarding exercise in binomial theorem application. It involves finding all the possible combinations of x, y, and z that add up to 5, accounting for their coefficients. Let's break down the process:
Understanding the Binomial Theorem
The binomial theorem provides a formula for expanding expressions of the form (a + b)^n:
(a + b)^n = ∑ (n choose k) * a^(n-k) * b^k
where:
- n is the power
- k ranges from 0 to n
- (n choose k) represents the binomial coefficient, calculated as n!/(k! * (n-k)!). This coefficient represents the number of ways to choose k items from a set of n items.
Applying the Theorem to (x + y + z)^5
While the binomial theorem directly addresses expressions with two terms, we can adapt it for three terms by considering (x + y + z) as a single entity:
- Identify the terms: We have three terms: x, y, and z.
- Determine the power: The power is 5.
- Expand the expression: We need to find all combinations of exponents for x, y, and z that sum up to 5. Each combination will have a corresponding coefficient calculated using the binomial theorem.
Example: One possible combination is x^3 * y^1 * z^1. Its coefficient is found by considering the binomial theorem applied to (x + (y+z))^5 and focusing on the term (y+z)^1:
- (5 choose 1) = 5 is the coefficient for (y+z)^1.
- (1 choose 1) = 1 is the coefficient for y^1 * z^1 within (y+z)^1.
Therefore, the coefficient for x^3 * y^1 * z^1 is 5 * 1 = 5.
The Complete Expansion
Following this process for all possible combinations, we arrive at the complete expansion of (x + y + z)^5:
(x + y + z)^5 = x^5 + 5x^4y + 5x^4z + 10x^3y^2 + 20x^3yz + 10x^3z^2 + 10x^2y^3 + 30x^2y^2z + 30x^2yz^2 + 10x^2z^3 + 5xy^4 + 20xy^3z + 30xy^2z^2 + 20xyz^3 + 5xz^4 + y^5 + 5y^4z + 10y^3z^2 + 10y^2z^3 + 5yz^4 + z^5
Key Points
- The number of terms in the expansion is (n+1)^3, where n is the power. For (x + y + z)^5, there are 6^3 = 216 terms.
- The coefficients in the expansion are the multinomial coefficients, which can be calculated using the formula: (n choose k1, k2, ..., kn) = n! / (k1! * k2! * ... * kn!), where k1 + k2 + ... + kn = n.
Applications
Understanding the expansion of (x + y + z)^5 has applications in various fields, including:
- Probability: In probability, expanding expressions like this helps determine the probability of various outcomes in events with multiple possibilities.
- Statistics: In statistics, it plays a role in understanding the distributions of variables and calculating moments.
- Algebraic manipulation: It provides a tool for manipulating and simplifying complex expressions.
This expansion serves as a building block for further mathematical exploration and problem-solving in various areas.