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Popular Algebra >

expand (x+1)^{20}

Solution

expand

(x + 1)^{20}

Solution

x^{20} + 20 x^{19} + 190 x^{18} + 1140 x^{17} + 4845 x^{16} + 15504 x^{15} + 38760 x^{14} + 77520 x^{13} + 125970 x^{12} + 167960 x^{11} + 184756 x^{10} + 167960 x^{9} + 125970 x^{8} + 77520 x^{7} + 38760 x^{6} + 15504 x^{5} + 4845 x^{4} + 1140 x^{3} + 190 x^{2} + 20 x + 1

Solution steps

(x + 1)^{20}

Apply binomial theorem:

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

a = x, b = 1

= i = 0 \sum 20 (i 20) x^{(20 - i)} \cdot 1^{i}

Expand summation

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

Simplify

\frac{20 !}{0 ! ( 20 - 0 ) !} x^{20} \cdot 1^{0} : x^{20}

Simplify

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

Simplify

\frac{20 !}{2 ! ( 20 - 2 ) !} x^{18} \cdot 1^{2} : 190 x^{18}

Simplify

\frac{20 !}{3 ! ( 20 - 3 ) !} x^{17} \cdot 1^{3} : 1140 x^{17}

Simplify

\frac{20 !}{4 ! ( 20 - 4 ) !} x^{16} \cdot 1^{4} : 4845 x^{16}

Simplify

\frac{20 !}{5 ! ( 20 - 5 ) !} x^{15} \cdot 1^{5} : 15504 x^{15}

Simplify

\frac{20 !}{6 ! ( 20 - 6 ) !} x^{14} \cdot 1^{6} : 38760 x^{14}

Simplify

\frac{20 !}{7 ! ( 20 - 7 ) !} x^{13} \cdot 1^{7} : 77520 x^{13}

Simplify

\frac{20 !}{8 ! ( 20 - 8 ) !} x^{12} \cdot 1^{8} : 125970 x^{12}

Simplify

\frac{20 !}{9 ! ( 20 - 9 ) !} x^{11} \cdot 1^{9} : 167960 x^{11}

Simplify

\frac{20 !}{10 ! ( 20 - 10 ) !} x^{10} \cdot 1^{10} : 184756 x^{10}

Simplify

\frac{20 !}{11 ! ( 20 - 11 ) !} x^{9} \cdot 1^{11} : 167960 x^{9}

Simplify

\frac{20 !}{12 ! ( 20 - 12 ) !} x^{8} \cdot 1^{12} : 125970 x^{8}

Simplify

\frac{20 !}{13 ! ( 20 - 13 ) !} x^{7} \cdot 1^{13} : 77520 x^{7}

Simplify

\frac{20 !}{14 ! ( 20 - 14 ) !} x^{6} \cdot 1^{14} : 38760 x^{6}

Simplify

\frac{20 !}{15 ! ( 20 - 15 ) !} x^{5} \cdot 1^{15} : 15504 x^{5}

Simplify

\frac{20 !}{16 ! ( 20 - 16 ) !} x^{4} \cdot 1^{16} : 4845 x^{4}

Simplify

\frac{20 !}{17 ! ( 20 - 17 ) !} x^{3} \cdot 1^{17} : 1140 x^{3}

Simplify

\frac{20 !}{18 ! ( 20 - 18 ) !} x^{2} \cdot 1^{18} : 190 x^{2}

Simplify

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

Simplify

\frac{20 !}{20 ! ( 20 - 20 ) !} x^{0} \cdot 1^{20} : 1

= x^{20} + 20 x^{19} + 190 x^{18} + 1140 x^{17} + 4845 x^{16} + 15504 x^{15} + 38760 x^{14} + 77520 x^{13} + 125970 x^{12} + 167960 x^{11} + 184756 x^{10} + 167960 x^{9} + 125970 x^{8} + 77520 x^{7} + 38760 x^{6} + 15504 x^{5} + 4845 x^{4} + 1140 x^{3} + 190 x^{2} + 20 x + 1

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