The binomial theorem is used to calculate the expansion (x + y)^{n} without carrying out direct multiplication. In the expansion x and y are real numbers and n is an integer.

For all positive integers n, the binomial (x + y) can be expanded:

(x + y)^{n }= x^{n }+ ^{n}C_{1} x^{(n-1)} and + ^{n}C_{2} x^{(n-2)} y^{2} + ^{n}C_{3} x^{(n-3)} y^{3} +… + And^{n}

Where the coefficients ^{n}C_{r} appearing on the **binomial expansion** they are called binomial coefficients; these can also be expressed as . To calculate them, the combinatorial formula is used:

^{n}C_{r} = n! / [(n - r)! ∙ r! ]

**Basic binomial expansions**

- (x + y)
^{0 }= x^{0 }= 1 - (x + y)
^{1}= x + {2! / [(2 - 1)! ∙ 1! ]} · X^{1}y = x + y - (x + y)
^{2 }= x^{2 }+ {2! / [(2-1)! ∙ 1! ]} · X^{1}and + {2! / [(2 - 2)! ∙ 2! ]} · X^{0}y^{2 }= x^{2 }+ 2xy + y^{2} - If n is a rational number and x is a real number such that | x | <1 then
- (1 + x)
^{n }= 1 + nx +^{n}C_{2}x^{2 }+ ⋯^{n}C_{r}x^{r }+ ⋯ ∞ - (1 + x)
^{-n }= 1 - nx +^{n}C_{2}x^{2 }+^{n}C_{3}x^{3 }+ ⋯ ∞ - (1 - x)
^{n }= 1 - nx +^{n}C_{2}x^{2 }+^{n}C_{3}x^{3 }+ ⋯ ∞ - (1 - x)
^{-n }= 1 + nx +^{n}C_{2}x^{2 }+ ⋯^{n}C_{r}x^{r }+ ⋯ ∞

- (1 + x)

**Examples**:

- Solve (x + 20)
^{3}.

Since n = 3, we have to:

(x + y)^{3 }= x^{3 }+3! / [(3 - 1)! ∙ 1!] · X^{2}and + 3! / [(3 - 2)! ∙ 2!] · X^{1}y^{2 }+ 3! / [(3 - 3)! ∙ 3!] · X^{0} y^{3}

= x^{3 }+ 3x^{2}y + 3xy^{2 }+ and^{3}

Now, as y = 20, we get

(x + 20)^{3 }= x^{3 }+ 3x^{2}(20) + 3x (20)^{2 }+ (20)^{3 }=

= x^{3 }+ 60x^{2 }+ 1200x + 8000

- Determine the binomial coefficients of the expansion (x + 2)
^{6}using the binomial theorem.

Comparing (x + 2)^{6} with (x + y)^{n}, we determine that y = 2 and n = 6. Therefore:

(x + y)^{6 }= x^{6 }+ 6! / [(6 - 1)! ∙ 1!] · X^{5}and + 6! / [(6 - 2)! ∙ 2!] · X^{4}y^{2} + 6! / [(6 - 3)! ∙ 3!] · X^{3}y^{3} + 6! / [(6 - 4)! ∙ 4!] · X^{2}y^{4} + 6! / [(6 - 5)! ∙ 5!] · X^{1}y^{5} + 6! / [(6 - 6)! ∙ 6!] · X^{0}y^{6 }=

= x^{6 }+ 6x^{2}y + 15x^{4}y^{2 }+ 20x^{3}y^{3 }+ 15x^{2}y^{4 }+ 6xy^{5 }+ and^{6}

Thus:

(x + 2)^{6} = x^{6 }+ 6x^{2}(2) + 15x^{4}(2)^{2 }+ 20x^{3}(2)^{3 }+ 15x^{2}(2)^{4 }+ 6x (2)^{5 }+ (2)^{6 }=

= x^{6 }+ 12x^{2} + 60x^{4 }+ 160x^{3 }+ 240x^{2 }+ 192x + 64

In consecuense:

- X coefficient
^{6 }= 1 - X coefficient
^{5 }= 12 - X coefficient
^{4 }= 60 - X coefficient
^{3 }= 160 - X coefficient
^{2 }= 240 - Coefficient of x = 192
- Constant = 64