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Popular Algebra >

expand (a+b)^{19}

Solution

expand

(a + b)^{19}

Solution

a^{19} + 19 a^{18} b + 171 a^{17} b^{2} + 969 a^{16} b^{3} + 3876 a^{15} b^{4} + 11628 a^{14} b^{5} + 27132 a^{13} b^{6} + 50388 a^{12} b^{7} + 75582 a^{11} b^{8} + 92378 a^{10} b^{9} + 92378 a^{9} b^{10} + 75582 a^{8} b^{11} + 50388 a^{7} b^{12} + 27132 a^{6} b^{13} + 11628 a^{5} b^{14} + 3876 a^{4} b^{15} + 969 a^{3} b^{16} + 171 a^{2} b^{17} + 19 a b^{18} + b^{19}

Solution steps

(a + b)^{19}

Apply binomial theorem:

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

a = a, b = b

= i = 0 \sum 19 (i 19) a^{(19 - i)} b^{i}

Expand summation

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

Simplify

\frac{19 !}{0 ! ( 19 - 0 ) !} a^{19} b^{0} : a^{19}

Simplify

\frac{19 !}{1 ! ( 19 - 1 ) !} a^{18} b^{1} : 19 a^{18} b

Simplify

\frac{19 !}{2 ! ( 19 - 2 ) !} a^{17} b^{2} : 171 a^{17} b^{2}

Simplify

\frac{19 !}{3 ! ( 19 - 3 ) !} a^{16} b^{3} : 969 a^{16} b^{3}

Simplify

\frac{19 !}{4 ! ( 19 - 4 ) !} a^{15} b^{4} : 3876 a^{15} b^{4}

Simplify

\frac{19 !}{5 ! ( 19 - 5 ) !} a^{14} b^{5} : 11628 a^{14} b^{5}

Simplify

\frac{19 !}{6 ! ( 19 - 6 ) !} a^{13} b^{6} : 27132 a^{13} b^{6}

Simplify

\frac{19 !}{7 ! ( 19 - 7 ) !} a^{12} b^{7} : 50388 a^{12} b^{7}

Simplify

\frac{19 !}{8 ! ( 19 - 8 ) !} a^{11} b^{8} : 75582 a^{11} b^{8}

Simplify

\frac{19 !}{9 ! ( 19 - 9 ) !} a^{10} b^{9} : 92378 a^{10} b^{9}

Simplify

\frac{19 !}{10 ! ( 19 - 10 ) !} a^{9} b^{10} : 92378 a^{9} b^{10}

Simplify

\frac{19 !}{11 ! ( 19 - 11 ) !} a^{8} b^{11} : 75582 a^{8} b^{11}

Simplify

\frac{19 !}{12 ! ( 19 - 12 ) !} a^{7} b^{12} : 50388 a^{7} b^{12}

Simplify

\frac{19 !}{13 ! ( 19 - 13 ) !} a^{6} b^{13} : 27132 a^{6} b^{13}

Simplify

\frac{19 !}{14 ! ( 19 - 14 ) !} a^{5} b^{14} : 11628 a^{5} b^{14}

Simplify

\frac{19 !}{15 ! ( 19 - 15 ) !} a^{4} b^{15} : 3876 a^{4} b^{15}

Simplify

\frac{19 !}{16 ! ( 19 - 16 ) !} a^{3} b^{16} : 969 a^{3} b^{16}

Simplify

\frac{19 !}{17 ! ( 19 - 17 ) !} a^{2} b^{17} : 171 a^{2} b^{17}

Simplify

\frac{19 !}{18 ! ( 19 - 18 ) !} a^{1} b^{18} : 19 a b^{18}

Simplify

\frac{19 !}{19 ! ( 19 - 19 ) !} a^{0} b^{19} : b^{19}

= a^{19} + 19 a^{18} b + 171 a^{17} b^{2} + 969 a^{16} b^{3} + 3876 a^{15} b^{4} + 11628 a^{14} b^{5} + 27132 a^{13} b^{6} + 50388 a^{12} b^{7} + 75582 a^{11} b^{8} + 92378 a^{10} b^{9} + 92378 a^{9} b^{10} + 75582 a^{8} b^{11} + 50388 a^{7} b^{12} + 27132 a^{6} b^{13} + 11628 a^{5} b^{14} + 3876 a^{4} b^{15} + 969 a^{3} b^{16} + 171 a^{2} b^{17} + 19 a b^{18} + b^{19}

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Frequently Asked Questions (FAQ)

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