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Popular >

expand (x+y)^{12}

Solution

expand

(x + y)^{12}

Solution

x^{12} + 12 x^{11} y + 66 x^{10} y^{2} + 220 x^{9} y^{3} + 495 x^{8} y^{4} + 792 x^{7} y^{5} + 924 x^{6} y^{6} + 792 x^{5} y^{7} + 495 x^{4} y^{8} + 220 x^{3} y^{9} + 66 x^{2} y^{10} + 12 x y^{11} + y^{12}

Solution steps

(x + y)^{12}

Apply binomial theorem:

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

a = x, b = y

= i = 0 \sum 12 (i 12) x^{(12 - i)} y^{i}

Expand summation

= \frac{12 !}{0 ! ( 12 - 0 ) !} x^{12} y^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} x^{11} y^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} x^{10} y^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} x^{9} y^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} x^{8} y^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} x^{7} y^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} x^{6} y^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} x^{5} y^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} x^{4} y^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} x^{3} y^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} x^{2} y^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} x^{1} y^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} x^{0} y^{12}

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} x^{12} y^{0} : x^{12}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} x^{11} y^{1} : 12 x^{11} y

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} x^{10} y^{2} : 66 x^{10} y^{2}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} x^{9} y^{3} : 220 x^{9} y^{3}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} x^{8} y^{4} : 495 x^{8} y^{4}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} x^{7} y^{5} : 792 x^{7} y^{5}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} x^{6} y^{6} : 924 x^{6} y^{6}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} x^{5} y^{7} : 792 x^{5} y^{7}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} x^{4} y^{8} : 495 x^{4} y^{8}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} x^{3} y^{9} : 220 x^{3} y^{9}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} x^{2} y^{10} : 66 x^{2} y^{10}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} x^{1} y^{11} : 12 x y^{11}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} x^{0} y^{12} : y^{12}

= x^{12} + 12 x^{11} y + 66 x^{10} y^{2} + 220 x^{9} y^{3} + 495 x^{8} y^{4} + 792 x^{7} y^{5} + 924 x^{6} y^{6} + 792 x^{5} y^{7} + 495 x^{4} y^{8} + 220 x^{3} y^{9} + 66 x^{2} y^{10} + 12 x y^{11} + y^{12}

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Frequently Asked Questions (FAQ)

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