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Popular Algebra >

expand (a+b)^{20}

Solution

expand

(a + b)^{20}

Solution

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

Solution steps

(a + b)^{20}

Apply binomial theorem:

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

a = a, b = b

= i = 0 \sum 20 (i 20) a^{(20 - i)} b^{i}

Expand summation

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

Simplify

\frac{20 !}{0 ! ( 20 - 0 ) !} a^{20} b^{0} : a^{20}

Simplify

\frac{20 !}{1 ! ( 20 - 1 ) !} a^{19} b^{1} : 20 a^{19} b

Simplify

\frac{20 !}{2 ! ( 20 - 2 ) !} a^{18} b^{2} : 190 a^{18} b^{2}

Simplify

\frac{20 !}{3 ! ( 20 - 3 ) !} a^{17} b^{3} : 1140 a^{17} b^{3}

Simplify

\frac{20 !}{4 ! ( 20 - 4 ) !} a^{16} b^{4} : 4845 a^{16} b^{4}

Simplify

\frac{20 !}{5 ! ( 20 - 5 ) !} a^{15} b^{5} : 15504 a^{15} b^{5}

Simplify

\frac{20 !}{6 ! ( 20 - 6 ) !} a^{14} b^{6} : 38760 a^{14} b^{6}

Simplify

\frac{20 !}{7 ! ( 20 - 7 ) !} a^{13} b^{7} : 77520 a^{13} b^{7}

Simplify

\frac{20 !}{8 ! ( 20 - 8 ) !} a^{12} b^{8} : 125970 a^{12} b^{8}

Simplify

\frac{20 !}{9 ! ( 20 - 9 ) !} a^{11} b^{9} : 167960 a^{11} b^{9}

Simplify

\frac{20 !}{10 ! ( 20 - 10 ) !} a^{10} b^{10} : 184756 a^{10} b^{10}

Simplify

\frac{20 !}{11 ! ( 20 - 11 ) !} a^{9} b^{11} : 167960 a^{9} b^{11}

Simplify

\frac{20 !}{12 ! ( 20 - 12 ) !} a^{8} b^{12} : 125970 a^{8} b^{12}

Simplify

\frac{20 !}{13 ! ( 20 - 13 ) !} a^{7} b^{13} : 77520 a^{7} b^{13}

Simplify

\frac{20 !}{14 ! ( 20 - 14 ) !} a^{6} b^{14} : 38760 a^{6} b^{14}

Simplify

\frac{20 !}{15 ! ( 20 - 15 ) !} a^{5} b^{15} : 15504 a^{5} b^{15}

Simplify

\frac{20 !}{16 ! ( 20 - 16 ) !} a^{4} b^{16} : 4845 a^{4} b^{16}

Simplify

\frac{20 !}{17 ! ( 20 - 17 ) !} a^{3} b^{17} : 1140 a^{3} b^{17}

Simplify

\frac{20 !}{18 ! ( 20 - 18 ) !} a^{2} b^{18} : 190 a^{2} b^{18}

Simplify

\frac{20 !}{19 ! ( 20 - 19 ) !} a^{1} b^{19} : 20 a b^{19}

Simplify

\frac{20 !}{20 ! ( 20 - 20 ) !} a^{0} b^{20} : b^{20}

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

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