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확장 (x/(20)+1)^{20}

해법

확장

(\frac{x}{20} + 1)^{20}

해법

\frac{x ^{20}}{2 0 ^{20}} + \frac{x ^{19}}{2 0 ^{18}} + \frac{19 x ^{18}}{2 ^{35} \cdot 5 ^{17}} + \frac{57 x ^{17}}{2 ^{32} \cdot 5 ^{16}} + \frac{969 x ^{16}}{5 ^{15} \cdot 4294967296} + \frac{969 x ^{15}}{5 ^{14} \cdot 335544320} + \frac{969 x ^{14}}{4.096 E 16} + \frac{969 x ^{13}}{1.024 E 15} + \frac{12597 x ^{12}}{4.096 E 14} + \frac{4199 x ^{11}}{5120000000000} + \frac{46189 x ^{10}}{2560000000000} + \frac{4199 x ^{9}}{12800000000} + \frac{12597 x ^{8}}{2560000000} + \frac{969 x ^{7}}{16000000} + \frac{969 x ^{6}}{1600000} + \frac{969 x ^{5}}{200000} + \frac{969 x ^{4}}{32000} + \frac{57 x ^{3}}{400} + \frac{19 x ^{2}}{40} + x + 1

솔루션 단계

(\frac{x}{20} + 1)^{20}

이항 정리 적용:

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

a = \frac{x}{20}, b = 1

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

합계 확장

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

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

간소화하다 :

\frac{x ^{20}}{2 0 ^{20}}

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

간소화하다 :

\frac{x ^{19}}{2 0 ^{18}}

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

간소화하다 :

\frac{19 x ^{18}}{2 ^{35} \cdot 5 ^{17}}

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

간소화하다 :

\frac{57 x ^{17}}{2 ^{32} \cdot 5 ^{16}}

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

간소화하다 :

\frac{969 x ^{16}}{5 ^{15} \cdot 4294967296}

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

간소화하다 :

\frac{969 x ^{15}}{5 ^{14} \cdot 335544320}

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

간소화하다 :

\frac{969 x ^{14}}{4.096 E 16}

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

간소화하다 :

\frac{969 x ^{13}}{1.024 E 15}

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

간소화하다 :

\frac{12597 x ^{12}}{4.096 E 14}

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

간소화하다 :

\frac{4199 x ^{11}}{5120000000000}

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

간소화하다 :

\frac{46189 x ^{10}}{2560000000000}

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

간소화하다 :

\frac{4199 x ^{9}}{12800000000}

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

간소화하다 :

\frac{12597 x ^{8}}{2560000000}

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

간소화하다 :

\frac{969 x ^{7}}{16000000}

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

간소화하다 :

\frac{969 x ^{6}}{1600000}

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

간소화하다 :

\frac{969 x ^{5}}{200000}

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

간소화하다 :

\frac{969 x ^{4}}{32000}

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

간소화하다 :

\frac{57 x ^{3}}{400}

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

간소화하다 :

\frac{19 x ^{2}}{40}

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

간소화하다 :

x

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

간소화하다 :

1

= \frac{x ^{20}}{2 0 ^{20}} + \frac{x ^{19}}{2 0 ^{18}} + \frac{19 x ^{18}}{2 ^{35} \cdot 5 ^{17}} + \frac{57 x ^{17}}{2 ^{32} \cdot 5 ^{16}} + \frac{969 x ^{16}}{5 ^{15} \cdot 4294967296} + \frac{969 x ^{15}}{5 ^{14} \cdot 335544320} + \frac{969 x ^{14}}{4.096 E 16} + \frac{969 x ^{13}}{1.024 E 15} + \frac{12597 x ^{12}}{4.096 E 14} + \frac{4199 x ^{11}}{5120000000000} + \frac{46189 x ^{10}}{2560000000000} + \frac{4199 x ^{9}}{12800000000} + \frac{12597 x ^{8}}{2560000000} + \frac{969 x ^{7}}{16000000} + \frac{969 x ^{6}}{1600000} + \frac{969 x ^{5}}{200000} + \frac{969 x ^{4}}{32000} + \frac{57 x ^{3}}{400} + \frac{19 x ^{2}}{40} + x + 1

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