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Popular Algebra >

expand (x+y)^{14}

Solution

expand

(x + y)^{14}

Solution

x^{14} + 14 x^{13} y + 91 x^{12} y^{2} + 364 x^{11} y^{3} + 1001 x^{10} y^{4} + 2002 x^{9} y^{5} + 3003 x^{8} y^{6} + 3432 x^{7} y^{7} + 3003 x^{6} y^{8} + 2002 x^{5} y^{9} + 1001 x^{4} y^{10} + 364 x^{3} y^{11} + 91 x^{2} y^{12} + 14 x y^{13} + y^{14}

Solution steps

(x + y)^{14}

Apply binomial theorem:

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

a = x, b = y

= i = 0 \sum 14 (i 14) x^{(14 - i)} y^{i}

Expand summation

= \frac{14 !}{0 ! ( 14 - 0 ) !} x^{14} y^{0} + \frac{14 !}{1 ! ( 14 - 1 ) !} x^{13} y^{1} + \frac{14 !}{2 ! ( 14 - 2 ) !} x^{12} y^{2} + \frac{14 !}{3 ! ( 14 - 3 ) !} x^{11} y^{3} + \frac{14 !}{4 ! ( 14 - 4 ) !} x^{10} y^{4} + \frac{14 !}{5 ! ( 14 - 5 ) !} x^{9} y^{5} + \frac{14 !}{6 ! ( 14 - 6 ) !} x^{8} y^{6} + \frac{14 !}{7 ! ( 14 - 7 ) !} x^{7} y^{7} + \frac{14 !}{8 ! ( 14 - 8 ) !} x^{6} y^{8} + \frac{14 !}{9 ! ( 14 - 9 ) !} x^{5} y^{9} + \frac{14 !}{10 ! ( 14 - 10 ) !} x^{4} y^{10} + \frac{14 !}{11 ! ( 14 - 11 ) !} x^{3} y^{11} + \frac{14 !}{12 ! ( 14 - 12 ) !} x^{2} y^{12} + \frac{14 !}{13 ! ( 14 - 13 ) !} x^{1} y^{13} + \frac{14 !}{14 ! ( 14 - 14 ) !} x^{0} y^{14}

Simplify

\frac{14 !}{0 ! ( 14 - 0 ) !} x^{14} y^{0} : x^{14}

Simplify

\frac{14 !}{1 ! ( 14 - 1 ) !} x^{13} y^{1} : 14 x^{13} y

Simplify

\frac{14 !}{2 ! ( 14 - 2 ) !} x^{12} y^{2} : 91 x^{12} y^{2}

Simplify

\frac{14 !}{3 ! ( 14 - 3 ) !} x^{11} y^{3} : 364 x^{11} y^{3}

Simplify

\frac{14 !}{4 ! ( 14 - 4 ) !} x^{10} y^{4} : 1001 x^{10} y^{4}

Simplify

\frac{14 !}{5 ! ( 14 - 5 ) !} x^{9} y^{5} : 2002 x^{9} y^{5}

Simplify

\frac{14 !}{6 ! ( 14 - 6 ) !} x^{8} y^{6} : 3003 x^{8} y^{6}

Simplify

\frac{14 !}{7 ! ( 14 - 7 ) !} x^{7} y^{7} : 3432 x^{7} y^{7}

Simplify

\frac{14 !}{8 ! ( 14 - 8 ) !} x^{6} y^{8} : 3003 x^{6} y^{8}

Simplify

\frac{14 !}{9 ! ( 14 - 9 ) !} x^{5} y^{9} : 2002 x^{5} y^{9}

Simplify

\frac{14 !}{10 ! ( 14 - 10 ) !} x^{4} y^{10} : 1001 x^{4} y^{10}

Simplify

\frac{14 !}{11 ! ( 14 - 11 ) !} x^{3} y^{11} : 364 x^{3} y^{11}

Simplify

\frac{14 !}{12 ! ( 14 - 12 ) !} x^{2} y^{12} : 91 x^{2} y^{12}

Simplify

\frac{14 !}{13 ! ( 14 - 13 ) !} x^{1} y^{13} : 14 x y^{13}

Simplify

\frac{14 !}{14 ! ( 14 - 14 ) !} x^{0} y^{14} : y^{14}

= x^{14} + 14 x^{13} y + 91 x^{12} y^{2} + 364 x^{11} y^{3} + 1001 x^{10} y^{4} + 2002 x^{9} y^{5} + 3003 x^{8} y^{6} + 3432 x^{7} y^{7} + 3003 x^{6} y^{8} + 2002 x^{5} y^{9} + 1001 x^{4} y^{10} + 364 x^{3} y^{11} + 91 x^{2} y^{12} + 14 x y^{13} + y^{14}

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