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Popular >

expand (a+b)^{12}

Solution

expand

(a + b)^{12}

Solution

a^{12} + 12 a^{11} b + 66 a^{10} b^{2} + 220 a^{9} b^{3} + 495 a^{8} b^{4} + 792 a^{7} b^{5} + 924 a^{6} b^{6} + 792 a^{5} b^{7} + 495 a^{4} b^{8} + 220 a^{3} b^{9} + 66 a^{2} b^{10} + 12 a b^{11} + b^{12}

Solution steps

(a + b)^{12}

Apply binomial theorem:

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

a = a, b = b

= i = 0 \sum 12 (i 12) a^{(12 - i)} b^{i}

Expand summation

= \frac{12 !}{0 ! ( 12 - 0 ) !} a^{12} b^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} a^{11} b^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} a^{10} b^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} a^{9} b^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} a^{8} b^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} a^{7} b^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} a^{6} b^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} a^{5} b^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} a^{4} b^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} a^{3} b^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} a^{2} b^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} a^{1} b^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} a^{0} b^{12}

Simplify

\frac{12 !}{0 ! ( 12 - 0 ) !} a^{12} b^{0} : a^{12}

Simplify

\frac{12 !}{1 ! ( 12 - 1 ) !} a^{11} b^{1} : 12 a^{11} b

Simplify

\frac{12 !}{2 ! ( 12 - 2 ) !} a^{10} b^{2} : 66 a^{10} b^{2}

Simplify

\frac{12 !}{3 ! ( 12 - 3 ) !} a^{9} b^{3} : 220 a^{9} b^{3}

Simplify

\frac{12 !}{4 ! ( 12 - 4 ) !} a^{8} b^{4} : 495 a^{8} b^{4}

Simplify

\frac{12 !}{5 ! ( 12 - 5 ) !} a^{7} b^{5} : 792 a^{7} b^{5}

Simplify

\frac{12 !}{6 ! ( 12 - 6 ) !} a^{6} b^{6} : 924 a^{6} b^{6}

Simplify

\frac{12 !}{7 ! ( 12 - 7 ) !} a^{5} b^{7} : 792 a^{5} b^{7}

Simplify

\frac{12 !}{8 ! ( 12 - 8 ) !} a^{4} b^{8} : 495 a^{4} b^{8}

Simplify

\frac{12 !}{9 ! ( 12 - 9 ) !} a^{3} b^{9} : 220 a^{3} b^{9}

Simplify

\frac{12 !}{10 ! ( 12 - 10 ) !} a^{2} b^{10} : 66 a^{2} b^{10}

Simplify

\frac{12 !}{11 ! ( 12 - 11 ) !} a^{1} b^{11} : 12 a b^{11}

Simplify

\frac{12 !}{12 ! ( 12 - 12 ) !} a^{0} b^{12} : b^{12}

= a^{12} + 12 a^{11} b + 66 a^{10} b^{2} + 220 a^{9} b^{3} + 495 a^{8} b^{4} + 792 a^{7} b^{5} + 924 a^{6} b^{6} + 792 a^{5} b^{7} + 495 a^{4} b^{8} + 220 a^{3} b^{9} + 66 a^{2} b^{10} + 12 a b^{11} + b^{12}

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Frequently Asked Questions (FAQ)

What is expand (a+b)^{12} ?
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