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受欢迎的代数 >

展开 (1/6+5/6)^5

解答

展开

(\frac{1}{6} + \frac{5}{6})^{5}

解答

1

求解步骤

(\frac{1}{6} + \frac{5}{6})^{5}

使用二项式定理:

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

a = \frac{1}{6}, b = \frac{5}{6}

= i = 0 \sum 5 (i 5) (\frac{1}{6})^{(5 - i)} (\frac{5}{6})^{i}

展开求和

= \frac{5 !}{0 ! ( 5 - 0 ) !} (\frac{1}{6})^{5} (\frac{5}{6})^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} (\frac{1}{6})^{4} (\frac{5}{6})^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} (\frac{1}{6})^{3} (\frac{5}{6})^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} (\frac{1}{6})^{2} (\frac{5}{6})^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} (\frac{1}{6})^{1} (\frac{5}{6})^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} (\frac{1}{6})^{0} (\frac{5}{6})^{5}

化简

\frac{5 !}{0 ! ( 5 - 0 ) !} (\frac{1}{6})^{5} (\frac{5}{6})^{0} : \frac{1}{7776}

化简

\frac{5 !}{1 ! ( 5 - 1 ) !} (\frac{1}{6})^{4} (\frac{5}{6})^{1} : \frac{25}{7776}

化简

\frac{5 !}{2 ! ( 5 - 2 ) !} (\frac{1}{6})^{3} (\frac{5}{6})^{2} : \frac{125}{3888}

化简

\frac{5 !}{3 ! ( 5 - 3 ) !} (\frac{1}{6})^{2} (\frac{5}{6})^{3} : \frac{625}{3888}

化简

\frac{5 !}{4 ! ( 5 - 4 ) !} (\frac{1}{6})^{1} (\frac{5}{6})^{4} : \frac{3125}{7776}

化简

\frac{5 !}{5 ! ( 5 - 5 ) !} (\frac{1}{6})^{0} (\frac{5}{6})^{5} : \frac{3125}{7776}

= \frac{1}{7776} + \frac{25}{7776} + \frac{125}{3888} + \frac{625}{3888} + \frac{3125}{7776} + \frac{3125}{7776}

\frac{1}{7776} + \frac{25}{7776} + \frac{125}{3888} + \frac{625}{3888} + \frac{3125}{7776} + \frac{3125}{7776} = 1

= 1

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