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Popular >

expand (1/2 p+1/q)^{18}

Solution

expand

(\frac{1}{2} p + \frac{1}{q})^{18}

Solution

\frac{p ^{18}}{262144} + \frac{9 p ^{17}}{65536 q} + \frac{153 p ^{16}}{65536 q ^{2}} + \frac{51 p ^{15}}{2048 q ^{3}} + \frac{765 p ^{14}}{4096 q ^{4}} + \frac{1071 p ^{13}}{1024 q ^{5}} + \frac{4641 p ^{12}}{1024 q ^{6}} + \frac{1989 p ^{11}}{128 q ^{7}} + \frac{21879 p ^{10}}{512 q ^{8}} + \frac{12155 p ^{9}}{128 q ^{9}} + \frac{21879 p ^{8}}{128 q ^{10}} + \frac{1989 p ^{7}}{8 q ^{11}} + \frac{4641 p ^{6}}{16 q ^{12}} + \frac{1071 p ^{5}}{4 q ^{13}} + \frac{765 p ^{4}}{4 q ^{14}} + \frac{102 p ^{3}}{q ^{15}} + \frac{153 p ^{2}}{4 q ^{16}} + \frac{9 p}{q ^{17}} + \frac{1}{q ^{18}}

Solution steps

(\frac{1}{2} p + \frac{1}{q})^{18}

Apply binomial theorem:

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

a = \frac{1}{2} p, b = \frac{1}{q}

= i = 0 \sum 18 (i 18) (\frac{1}{2} p)^{(18 - i)} (\frac{1}{q})^{i}

Expand summation

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

Simplify

\frac{18 !}{0 ! ( 18 - 0 ) !} (\frac{1}{2} p)^{18} (\frac{1}{q})^{0} : \frac{p ^{18}}{262144}

Simplify

\frac{18 !}{1 ! ( 18 - 1 ) !} (\frac{1}{2} p)^{17} (\frac{1}{q})^{1} : \frac{9 p ^{17}}{65536 q}

Simplify

\frac{18 !}{2 ! ( 18 - 2 ) !} (\frac{1}{2} p)^{16} (\frac{1}{q})^{2} : \frac{153 p ^{16}}{65536 q ^{2}}

Simplify

\frac{18 !}{3 ! ( 18 - 3 ) !} (\frac{1}{2} p)^{15} (\frac{1}{q})^{3} : \frac{51 p ^{15}}{2048 q ^{3}}

Simplify

\frac{18 !}{4 ! ( 18 - 4 ) !} (\frac{1}{2} p)^{14} (\frac{1}{q})^{4} : \frac{765 p ^{14}}{4096 q ^{4}}

Simplify

\frac{18 !}{5 ! ( 18 - 5 ) !} (\frac{1}{2} p)^{13} (\frac{1}{q})^{5} : \frac{1071 p ^{13}}{1024 q ^{5}}

Simplify

\frac{18 !}{6 ! ( 18 - 6 ) !} (\frac{1}{2} p)^{12} (\frac{1}{q})^{6} : \frac{4641 p ^{12}}{1024 q ^{6}}

Simplify

\frac{18 !}{7 ! ( 18 - 7 ) !} (\frac{1}{2} p)^{11} (\frac{1}{q})^{7} : \frac{1989 p ^{11}}{128 q ^{7}}

Simplify

\frac{18 !}{8 ! ( 18 - 8 ) !} (\frac{1}{2} p)^{10} (\frac{1}{q})^{8} : \frac{21879 p ^{10}}{512 q ^{8}}

Simplify

\frac{18 !}{9 ! ( 18 - 9 ) !} (\frac{1}{2} p)^{9} (\frac{1}{q})^{9} : \frac{12155 p ^{9}}{128 q ^{9}}

Simplify

\frac{18 !}{10 ! ( 18 - 10 ) !} (\frac{1}{2} p)^{8} (\frac{1}{q})^{10} : \frac{21879 p ^{8}}{128 q ^{10}}

Simplify

\frac{18 !}{11 ! ( 18 - 11 ) !} (\frac{1}{2} p)^{7} (\frac{1}{q})^{11} : \frac{1989 p ^{7}}{8 q ^{11}}

Simplify

\frac{18 !}{12 ! ( 18 - 12 ) !} (\frac{1}{2} p)^{6} (\frac{1}{q})^{12} : \frac{4641 p ^{6}}{16 q ^{12}}

Simplify

\frac{18 !}{13 ! ( 18 - 13 ) !} (\frac{1}{2} p)^{5} (\frac{1}{q})^{13} : \frac{1071 p ^{5}}{4 q ^{13}}

Simplify

\frac{18 !}{14 ! ( 18 - 14 ) !} (\frac{1}{2} p)^{4} (\frac{1}{q})^{14} : \frac{765 p ^{4}}{4 q ^{14}}

Simplify

\frac{18 !}{15 ! ( 18 - 15 ) !} (\frac{1}{2} p)^{3} (\frac{1}{q})^{15} : \frac{102 p ^{3}}{q ^{15}}

Simplify

\frac{18 !}{16 ! ( 18 - 16 ) !} (\frac{1}{2} p)^{2} (\frac{1}{q})^{16} : \frac{153 p ^{2}}{4 q ^{16}}

Simplify

\frac{18 !}{17 ! ( 18 - 17 ) !} (\frac{1}{2} p)^{1} (\frac{1}{q})^{17} : \frac{9 p}{q ^{17}}

Simplify

\frac{18 !}{18 ! ( 18 - 18 ) !} (\frac{1}{2} p)^{0} (\frac{1}{q})^{18} : \frac{1}{q ^{18}}

= \frac{p ^{18}}{262144} + \frac{9 p ^{17}}{65536 q} + \frac{153 p ^{16}}{65536 q ^{2}} + \frac{51 p ^{15}}{2048 q ^{3}} + \frac{765 p ^{14}}{4096 q ^{4}} + \frac{1071 p ^{13}}{1024 q ^{5}} + \frac{4641 p ^{12}}{1024 q ^{6}} + \frac{1989 p ^{11}}{128 q ^{7}} + \frac{21879 p ^{10}}{512 q ^{8}} + \frac{12155 p ^{9}}{128 q ^{9}} + \frac{21879 p ^{8}}{128 q ^{10}} + \frac{1989 p ^{7}}{8 q ^{11}} + \frac{4641 p ^{6}}{16 q ^{12}} + \frac{1071 p ^{5}}{4 q ^{13}} + \frac{765 p ^{4}}{4 q ^{14}} + \frac{102 p ^{3}}{q ^{15}} + \frac{153 p ^{2}}{4 q ^{16}} + \frac{9 p}{q ^{17}} + \frac{1}{q ^{18}}

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Frequently Asked Questions (FAQ)

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