More Pascal’s triangle expansions (HSC SSCE Mathematics Extension 1): Flashcards
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What is Pascal's Triangle?
What is Pascal's Triangle?
A triangle pattern with rows starting and ending in '1'.
How is each middle number in Pascal's Triangle calculated?
How is each middle number in Pascal's Triangle calculated?
It's the sum of the two numbers directly above it.
What does binomial expansion do?
What does binomial expansion do?
It expands powers of a binomial expression.
What is the formula for binomial coefficients?
What is the formula for binomial coefficients?
nCr = n! / (r!(n-r)!)
How do you build the 5th row of Pascal's Triangle?
How do you build the 5th row of Pascal's Triangle?
Start with '1', then add numbers from the 4th row.
What is the sum of each row in Pascal's Triangle?
What is the sum of each row in Pascal's Triangle?
Each row's sum equals 2^n.
How do you expand (2 + x)^3 using Pascal's Triangle?
How do you expand (2 + x)^3 using Pascal's Triangle?
Use row 3: 1, 3, 3, 1 for coefficients.
What is the result of expanding (3 - x)^3?
What is the result of expanding (3 - x)^3?
Result: 27 - 27x + 9x^2 - x^3.
What is a key property of binomial coefficients?
What is a key property of binomial coefficients?
They exhibit symmetry: nCr = nC(n-r).
How do you calculate nCr for large n?
How do you calculate nCr for large n?
Use the factorial formula for efficiency.