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{9 x ^{4} y ^{5}}{3 x ^{8} y ^{3}}

\frac{25}{40}

3 \times \frac{3}{5}

ln (\frac{1}{e ^{y}})

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