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OCR A-level Maths Binomial expansion

Binomial expansion

This page covers the following topics:

A factorial is the product of all whole numbers smaller than or equal to a given number. The symbol of the factorial function is !. Factorials can be calculated on a calculator by pressing the number followed by the factorial button (!).

Factorials

Pascal's triangle is a triangular pattern of numbers, beginning with a 1 on top, that are arranged so that each number is the sum of the two numbers directly above it. It can be used to determine the coefficients of a binomial expansion. When expanding a binomial of the type (x+y)n, the coefficients correspond to the numbers in the nth row of Pascal?s triangle (counting the top row, consisting of a 1, as the 0th row).

Pascal's triangle

For brackets with larger powers, it can be impractical to use Pascal's Triangle to determine the coefficients. Instead, they can be calculated by the formula NC(r) (see image). There is a button on the calculator for this. If n is a positive integer, the bracket can then be expanded using the binomial theorem: (a + b)n = an + (nC1)anβˆ’1b + (nC2)anβˆ’2b2 + … + (nCnβˆ’1)abnβˆ’1 + bn.

Binomial expansion

1

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Calculate 8!

OCR A-level Maths Binomial expansion Calculate 8!
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2

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Calculate 10!.

OCR A-level Maths Binomial expansion Calculate 10!.
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3

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Which row of Pascal's triangle contains the coefficients of the expansion of (3x + 2)Β²? What are the coefficients?

OCR A-level Maths Binomial expansion Which row of Pascal's triangle contains the coefficients of the expansion of (3x + 2)Β²? What are the coefficients?
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4

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Find the coefficient of the second term of the expansion of (5x + 5)Β².

OCR A-level Maths Binomial expansion Find the coefficient of the second term of the expansion of (5x + 5)Β².
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5

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What is the coefficient of the third term of the expansion of (x + 4)Β³?

OCR A-level Maths Binomial expansion What is the coefficient of the third term of the expansion of (x + 4)Β³?
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6

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Find the coefficient of the term that has x²y⁴ after expanding (5x + 3y)⁢.

OCR A-level Maths Binomial expansion Find the coefficient of the term that has x²y⁴ after expanding (5x + 3y)⁢.
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7

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Find the coefficient of the term that has aΒ²bΒ² after expanding (2a + 3bΒ²)Β³.

OCR A-level Maths Binomial expansion Find the coefficient of the term that has aΒ²bΒ² after expanding (2a + 3bΒ²)Β³.
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8

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Expand (2x βˆ’ 3)⁡.

OCR A-level Maths Binomial expansion Expand (2x βˆ’ 3)⁡.
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