Binomial Expansion
Pascal’s Triangle
Courtesy of Wikipedia
● You can expand an expression in the form (a + b)n using Pascal’s triangle
● Find the n th row of the triangle
● The coefficient of each term is multiplied by the corresponding number on
the row of the triangle (ie the second term’s coefficient is multiplied by the
second number in the appropriate row of the triangle)
● The first term is an and the powers of a decrease by 1 each term (so the
final term has no a in it)
● The first term has no b in it and the powers of b increase by 1 each term (so
the final term is bn
Combinations
● For large values of n , using Pascal’s triangle is impractical
● Instead use nCr to
give the coefficient multiplication numbers
n!
r!(n−r)!
● n is the power to which the entire bracket is raised, r is the power to which
the x of the desired term is raised
● nCr is the same as saying “How many ways are there of taking r items from
a collection of n items”
● P is called Permutations and is the same except is used when order
n r
matters (ie if different orders are different unique entities) and so will give a
higher number than nCr
Approximations
● Can be used to make approximations of decimals raised to certain powers
● For example 1.0230 is the same as (1 + 2x)30 where x is equal to 0.01
● Therefore to approximate, use binomial expansion to find the first three or
four terms and substitute x = 0.01
Pascal’s Triangle
Courtesy of Wikipedia
● You can expand an expression in the form (a + b)n using Pascal’s triangle
● Find the n th row of the triangle
● The coefficient of each term is multiplied by the corresponding number on
the row of the triangle (ie the second term’s coefficient is multiplied by the
second number in the appropriate row of the triangle)
● The first term is an and the powers of a decrease by 1 each term (so the
final term has no a in it)
● The first term has no b in it and the powers of b increase by 1 each term (so
the final term is bn
Combinations
● For large values of n , using Pascal’s triangle is impractical
● Instead use nCr to
give the coefficient multiplication numbers
n!
r!(n−r)!
● n is the power to which the entire bracket is raised, r is the power to which
the x of the desired term is raised
● nCr is the same as saying “How many ways are there of taking r items from
a collection of n items”
● P is called Permutations and is the same except is used when order
n r
matters (ie if different orders are different unique entities) and so will give a
higher number than nCr
Approximations
● Can be used to make approximations of decimals raised to certain powers
● For example 1.0230 is the same as (1 + 2x)30 where x is equal to 0.01
● Therefore to approximate, use binomial expansion to find the first three or
four terms and substitute x = 0.01
