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Popular Algebra >

expand (a+b)^{11}

Solution

expand

(a + b)^{11}

Solution

a^{11} + 11 a^{10} b + 55 a^{9} b^{2} + 165 a^{8} b^{3} + 330 a^{7} b^{4} + 462 a^{6} b^{5} + 462 a^{5} b^{6} + 330 a^{4} b^{7} + 165 a^{3} b^{8} + 55 a^{2} b^{9} + 11 a b^{10} + b^{11}

Solution steps

(a + b)^{11}

Apply binomial theorem:

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

a = a, b = b

= i = 0 \sum 11 (i 11) a^{(11 - i)} b^{i}

Expand summation

= \frac{11 !}{0 ! ( 11 - 0 ) !} a^{11} b^{0} + \frac{11 !}{1 ! ( 11 - 1 ) !} a^{10} b^{1} + \frac{11 !}{2 ! ( 11 - 2 ) !} a^{9} b^{2} + \frac{11 !}{3 ! ( 11 - 3 ) !} a^{8} b^{3} + \frac{11 !}{4 ! ( 11 - 4 ) !} a^{7} b^{4} + \frac{11 !}{5 ! ( 11 - 5 ) !} a^{6} b^{5} + \frac{11 !}{6 ! ( 11 - 6 ) !} a^{5} b^{6} + \frac{11 !}{7 ! ( 11 - 7 ) !} a^{4} b^{7} + \frac{11 !}{8 ! ( 11 - 8 ) !} a^{3} b^{8} + \frac{11 !}{9 ! ( 11 - 9 ) !} a^{2} b^{9} + \frac{11 !}{10 ! ( 11 - 10 ) !} a^{1} b^{10} + \frac{11 !}{11 ! ( 11 - 11 ) !} a^{0} b^{11}

Simplify

\frac{11 !}{0 ! ( 11 - 0 ) !} a^{11} b^{0} : a^{11}

Simplify

\frac{11 !}{1 ! ( 11 - 1 ) !} a^{10} b^{1} : 11 a^{10} b

Simplify

\frac{11 !}{2 ! ( 11 - 2 ) !} a^{9} b^{2} : 55 a^{9} b^{2}

Simplify

\frac{11 !}{3 ! ( 11 - 3 ) !} a^{8} b^{3} : 165 a^{8} b^{3}

Simplify

\frac{11 !}{4 ! ( 11 - 4 ) !} a^{7} b^{4} : 330 a^{7} b^{4}

Simplify

\frac{11 !}{5 ! ( 11 - 5 ) !} a^{6} b^{5} : 462 a^{6} b^{5}

Simplify

\frac{11 !}{6 ! ( 11 - 6 ) !} a^{5} b^{6} : 462 a^{5} b^{6}

Simplify

\frac{11 !}{7 ! ( 11 - 7 ) !} a^{4} b^{7} : 330 a^{4} b^{7}

Simplify

\frac{11 !}{8 ! ( 11 - 8 ) !} a^{3} b^{8} : 165 a^{3} b^{8}

Simplify

\frac{11 !}{9 ! ( 11 - 9 ) !} a^{2} b^{9} : 55 a^{2} b^{9}

Simplify

\frac{11 !}{10 ! ( 11 - 10 ) !} a^{1} b^{10} : 11 a b^{10}

Simplify

\frac{11 !}{11 ! ( 11 - 11 ) !} a^{0} b^{11} : b^{11}

= a^{11} + 11 a^{10} b + 55 a^{9} b^{2} + 165 a^{8} b^{3} + 330 a^{7} b^{4} + 462 a^{6} b^{5} + 462 a^{5} b^{6} + 330 a^{4} b^{7} + 165 a^{3} b^{8} + 55 a^{2} b^{9} + 11 a b^{10} + b^{11}

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