26 ## MULTIPLICATION OF SUMS## A proof of the binomial theoremTHE BINOMIAL THEOREM gives the coefficient of each term in the product of ( If we actually multiplied the 4 factors of ( we would find terms in
The binomial coefficients are 1 4 6 4 1 There is but 1 term in The binomial theorem is that those coefficients are the combinatorial numbers. To prove that, we will first consider the multiplication of any sums; for example: ( Upon multiplying, we would find six terms. Each term will contain two factors, namely one letter from each factor:
Therefore, we can write the product of the following— ( —simply by writing the sum of all combinations of one letter from each factor.
Each term in the product consists of three factors: one from each binomial. Note that there are a total of 2³ or 8 terms. In general: Multiplication of n binomials produces 2 For multiplication of two binomials gives 4 terms: ( If we multiply those with a binomial, we will have 8 terms; those multiplied with a binomial will produce 16 terms; and so on.
Example. (
How is the coefficient of each power of Each term in the product will have 4 factors, one from each binomial.
Terms with
The coefficient of
Next, terms with (
A term in ( There will be Finally, the constant term will be produced by taking the letter from each of the 4 binomials. There is The binomial theorem If, in the four binomials above, we make all the letters equal, then we have ( Each coefficient of a power of (
Now, the binomial coefficients are how many terms of each kind. We saw that the number of terms with The number of terms with The number of The number of The number of In other words:
The binomial coefficients are the combinatorial numbers. This can be generalized for any exponent
Compare Example 4, Lesson 25.
Problem 1. Imagine multiplying out ( a) How many terms would there be?
3 b) Each term would consist of how many factors? Two Problem 2. Write the product by taking the correct combinations of the integers. a) ( b) ( c) (
Problem 3. In this multiplication ( Problem 4. ( a) Upon multiplying out, and before collecting like terms, how many
b) How will a term
By taking c) How many
The number of ways of choosing 2 things—letter
Problem 5. In each row of Pascal's triangle, the sum of the binomial coefficients is 2
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