What is expand (x+y)^7 ?

The solution to expand (x+y)^7 is x^7+7x^6y+21x^5y^2+35x^4y^3+35x^3y^4+21x^2y^5+7xy^6+y^7

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expand (x+y)^7

expand

(x + y)^{7}

x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7}

Solution steps

(x + y)^{7}

Apply binomial theorem:

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

a = x, b = y

= i = 0 \sum 7 (i 7) x^{(7 - i)} y^{i}

Expand summation

= \frac{7 !}{0 ! ( 7 - 0 ) !} x^{7} y^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} x^{6} y^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} x^{5} y^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} x^{4} y^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} x^{3} y^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} x^{2} y^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} x^{1} y^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} x^{0} y^{7}

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} x^{7} y^{0} : x^{7}

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} x^{6} y^{1} : 7 x^{6} y

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} x^{5} y^{2} : 21 x^{5} y^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} x^{4} y^{3} : 35 x^{4} y^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} x^{3} y^{4} : 35 x^{3} y^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} x^{2} y^{5} : 21 x^{2} y^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} x^{1} y^{6} : 7 x y^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} x^{0} y^{7} : y^{7}

= x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7}

- 3 \cdot (- 2)

4 \frac{2}{3}

\frac{1}{e}

3 (m - 4)

\frac{6}{30}

What is expand (x+y)^7 ?
The solution to expand (x+y)^7 is x^7+7x^6y+21x^5y^2+35x^4y^3+35x^3y^4+21x^2y^5+7xy^6+y^7