erweitern (1/6+5/6)^5

Lösung

erweitern

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

1

Schritte zur Lösung

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

Wende Binomialsetz an:

(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}

Erweitere Summation

= \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}

Vereinfache

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

Vereinfache

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

Vereinfache

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

Vereinfache

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

Vereinfache

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

Vereinfache

\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

e x p an d c (x + 3 y - \frac{2 a}{3}) (x - 3 y - \frac{2 a}{3}) (x + \frac{a}{3})

e x p an d \frac{1}{4} (- 3 (3 + 2 x) + 11 ln (3 + 2 x)) + c

s im pl i f y (x^{2} - 9) (x + 5)

e x p an d - 6 e^{3 x} x + 3 e^{3 x} (- 3 x^{2} + 2)

e x p an d - \frac{2 m ( 2 m - n )}{2 m + n}