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The Binomial Theorem

The binomial theorem gives the coefficients of the expansion of powers of binomial expressions
Teorema binomial memberikan koefisien ekspansi pangkat ekspresi binomial
A binomial expression is simply the sum of two terms, such as x+yx + y.
Ekspresi binomial merupakan jumlah dari dua bentuk, seperti x+yx + y.

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Example 1

The expansion of (x+y)3(x + y)^3 can be found using combinatorial reasoning instead of multiplying the three terms out.
When (x+y)3=(x+y)(x+y)(x+y)(x + y)^3 = (x + y)(x + y)(x + y) is expanded, all products of a term in the first sum, a term in the second sum, and a term in the third sum are added.
Terms of the form x3 ,x2y ,xy2x^3~, x^2y~, xy^2, and y3y^3 arise.
To obtain a term of the form x3x^3, an x must be chosen in each of the sums, and this can be done in only one way. Thus, the x3x^3 term in the product has a coefficient of 1
It continues for x2y,xy2x^2y, xy^2, and y3y^3 and it follows that
Binomial Coefficients
Let x and y be variables, and let n be a nonnegative integer. Then
Binomial

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Example 2

What is the expansion of (x+y)4(x + y)^4?Solution:
From the binomial theorem it follows that
(x+y)4=∑j=04(4j)x4−jyj(x + y)^4 = \sum_{j=0}^{4} \binom{4}{j} x^{4-j} y^j
               =(40)x4+(41)x3y+(42)x2y2+(43)xy3+(44)y4~~~~~~~~~~~~~~~= \binom{4}{0} x^4 + \binom{4}{1} x^3 y + \binom{4}{2} x^2 y^2 + \binom{4}{3} x y^3 + \binom{4}{4} y^4
               =x4+4x3y+6x2y2+4xy3+y4~~~~~~~~~~~~~~~= x^4 + 4x^3 y + 6x^2 y^2 + 4x y^3 + y^4

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Example 3

What is the coefficient of x12y13x^{12} y^{13} in the expansion of (x+y)25(x + y)^25?Solution:
From the binomial theorem it follows that this coefficient is
(2513)=25!13! 12!=5 200 300\binom{25}{13} = \dfrac{25!}{13!~12!} = 5~200~300

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Corollary 1

Let nn be a nonnegative integer. Then
∑k=0n(nk)=2n\sum_{k = 0}^{n} \binom{n}{k} = 2^n