What is expand (x+y)^6 ?

The solution to expand (x+y)^6 is x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6

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expand (x+y)^6

expand

(x + y)^{6}

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

Solution steps

(x + y)^{6}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = y

= i = 0 \sum 6 (i 6) x^{(6 - i)} y^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} y^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} y^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} y^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} y^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} y^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} y^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} y^{6}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} y^{0} : x^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} y^{1} : 6 x^{5} y

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} y^{2} : 15 x^{4} y^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} y^{3} : 20 x^{3} y^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} y^{4} : 15 x^{2} y^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} y^{5} : 6 x y^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} y^{6} : y^{6}

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

\frac{8 x ^{6} - 10 x ^{4} - 12 x ^{3}}{- 4 x ^{2}}

- x^{2} - 4

\frac{225}{180}

\frac{1}{2} sin (2 x)

(7 - i) (1 + 5 i)

What is expand (x+y)^6 ?
The solution to expand (x+y)^6 is x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6