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Counting & Binomials
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The binomial theorem allows us expand expressions of the form (a+b)n:
or in summation form:
Example
The expansion of (3+x)4 is
Problems often ask you to find the coefficient of a specific term in a binomial expansion. The coefficient is the number multiplying a specific power of x. For example, in the expansion
the coefficient of x2 is 1, the coefficient of x is 2, and the coefficient of x0 is 1.
To find a specifically requested coefficient, remember that each term is of the form
for some r=0…n.
For example, to find the coefficient of x4 in the expansion of (x2−x2)8, we note that the general term is
where a is the coefficient to be determined. Focus on the powers of x, ignoring all constants:
So 16−3r=4⇒r=4. Plugging this back into the general term:
gives
(using a calculator)
Factorials are shortcut used to express decreasing products of integers such as
The definition is
Alternatively, n! can be defined recursively by:
This second definition is helpful when simplifying fractions of factorials:
Example
Find 8!11!.
The number in the nth row and rth column of Pascal's triangle (the rows and columns start at zero) is denoted by nCr, where 0≤r≤n. Alternative notations include
This number can be calculated using the formula
Notice, either using the formula or Pascal's Triangle that
and
Example
Pascal's triangle is a triangular array where each number is the sum of the two directly above it, beautifully revealing the coefficients of binomial expansions. Its symmetry and simple construction make it a powerful tool for exploring combinatorial relationships and probability.
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