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Popular Algebra >

expand (x-y)^{20}

Solution

expand

(x - y)^{20}

Solution

x^{20} - 20 x^{19} y + 190 x^{18} y^{2} - 1140 x^{17} y^{3} + 4845 x^{16} y^{4} - 15504 x^{15} y^{5} + 38760 x^{14} y^{6} - 77520 x^{13} y^{7} + 125970 x^{12} y^{8} - 167960 x^{11} y^{9} + 184756 x^{10} y^{10} - 167960 x^{9} y^{11} + 125970 x^{8} y^{12} - 77520 x^{7} y^{13} + 38760 x^{6} y^{14} - 15504 x^{5} y^{15} + 4845 x^{4} y^{16} - 1140 x^{3} y^{17} + 190 x^{2} y^{18} - 20 x y^{19} + y^{20}

Solution steps

(x - y)^{20}

Apply binomial theorem:

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

a = x, b = - y

= i = 0 \sum 20 (i 20) x^{(20 - i)} (- y)^{i}

Expand summation

= \frac{20 !}{0 ! ( 20 - 0 ) !} x^{20} (- y)^{0} + \frac{20 !}{1 ! ( 20 - 1 ) !} x^{19} (- y)^{1} + \frac{20 !}{2 ! ( 20 - 2 ) !} x^{18} (- y)^{2} + \frac{20 !}{3 ! ( 20 - 3 ) !} x^{17} (- y)^{3} + \frac{20 !}{4 ! ( 20 - 4 ) !} x^{16} (- y)^{4} + \frac{20 !}{5 ! ( 20 - 5 ) !} x^{15} (- y)^{5} + \frac{20 !}{6 ! ( 20 - 6 ) !} x^{14} (- y)^{6} + \frac{20 !}{7 ! ( 20 - 7 ) !} x^{13} (- y)^{7} + \frac{20 !}{8 ! ( 20 - 8 ) !} x^{12} (- y)^{8} + \frac{20 !}{9 ! ( 20 - 9 ) !} x^{11} (- y)^{9} + \frac{20 !}{10 ! ( 20 - 10 ) !} x^{10} (- y)^{10} + \frac{20 !}{11 ! ( 20 - 11 ) !} x^{9} (- y)^{11} + \frac{20 !}{12 ! ( 20 - 12 ) !} x^{8} (- y)^{12} + \frac{20 !}{13 ! ( 20 - 13 ) !} x^{7} (- y)^{13} + \frac{20 !}{14 ! ( 20 - 14 ) !} x^{6} (- y)^{14} + \frac{20 !}{15 ! ( 20 - 15 ) !} x^{5} (- y)^{15} + \frac{20 !}{16 ! ( 20 - 16 ) !} x^{4} (- y)^{16} + \frac{20 !}{17 ! ( 20 - 17 ) !} x^{3} (- y)^{17} + \frac{20 !}{18 ! ( 20 - 18 ) !} x^{2} (- y)^{18} + \frac{20 !}{19 ! ( 20 - 19 ) !} x^{1} (- y)^{19} + \frac{20 !}{20 ! ( 20 - 20 ) !} x^{0} (- y)^{20}

Simplify

\frac{20 !}{0 ! ( 20 - 0 ) !} x^{20} (- y)^{0} : x^{20}

Simplify

\frac{20 !}{1 ! ( 20 - 1 ) !} x^{19} (- y)^{1} : - 20 x^{19} y

Simplify

\frac{20 !}{2 ! ( 20 - 2 ) !} x^{18} (- y)^{2} : 190 x^{18} y^{2}

Simplify

\frac{20 !}{3 ! ( 20 - 3 ) !} x^{17} (- y)^{3} : - 1140 x^{17} y^{3}

Simplify

\frac{20 !}{4 ! ( 20 - 4 ) !} x^{16} (- y)^{4} : 4845 x^{16} y^{4}

Simplify

\frac{20 !}{5 ! ( 20 - 5 ) !} x^{15} (- y)^{5} : - 15504 x^{15} y^{5}

Simplify

\frac{20 !}{6 ! ( 20 - 6 ) !} x^{14} (- y)^{6} : 38760 x^{14} y^{6}

Simplify

\frac{20 !}{7 ! ( 20 - 7 ) !} x^{13} (- y)^{7} : - 77520 x^{13} y^{7}

Simplify

\frac{20 !}{8 ! ( 20 - 8 ) !} x^{12} (- y)^{8} : 125970 x^{12} y^{8}

Simplify

\frac{20 !}{9 ! ( 20 - 9 ) !} x^{11} (- y)^{9} : - 167960 x^{11} y^{9}

Simplify

\frac{20 !}{10 ! ( 20 - 10 ) !} x^{10} (- y)^{10} : 184756 x^{10} y^{10}

Simplify

\frac{20 !}{11 ! ( 20 - 11 ) !} x^{9} (- y)^{11} : - 167960 x^{9} y^{11}

Simplify

\frac{20 !}{12 ! ( 20 - 12 ) !} x^{8} (- y)^{12} : 125970 x^{8} y^{12}

Simplify

\frac{20 !}{13 ! ( 20 - 13 ) !} x^{7} (- y)^{13} : - 77520 x^{7} y^{13}

Simplify

\frac{20 !}{14 ! ( 20 - 14 ) !} x^{6} (- y)^{14} : 38760 x^{6} y^{14}

Simplify

\frac{20 !}{15 ! ( 20 - 15 ) !} x^{5} (- y)^{15} : - 15504 x^{5} y^{15}

Simplify

\frac{20 !}{16 ! ( 20 - 16 ) !} x^{4} (- y)^{16} : 4845 x^{4} y^{16}

Simplify

\frac{20 !}{17 ! ( 20 - 17 ) !} x^{3} (- y)^{17} : - 1140 x^{3} y^{17}

Simplify

\frac{20 !}{18 ! ( 20 - 18 ) !} x^{2} (- y)^{18} : 190 x^{2} y^{18}

Simplify

\frac{20 !}{19 ! ( 20 - 19 ) !} x^{1} (- y)^{19} : - 20 x y^{19}

Simplify

\frac{20 !}{20 ! ( 20 - 20 ) !} x^{0} (- y)^{20} : y^{20}

= x^{20} - 20 x^{19} y + 190 x^{18} y^{2} - 1140 x^{17} y^{3} + 4845 x^{16} y^{4} - 15504 x^{15} y^{5} + 38760 x^{14} y^{6} - 77520 x^{13} y^{7} + 125970 x^{12} y^{8} - 167960 x^{11} y^{9} + 184756 x^{10} y^{10} - 167960 x^{9} y^{11} + 125970 x^{8} y^{12} - 77520 x^{7} y^{13} + 38760 x^{6} y^{14} - 15504 x^{5} y^{15} + 4845 x^{4} y^{16} - 1140 x^{3} y^{17} + 190 x^{2} y^{18} - 20 x y^{19} + y^{20}

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