What is expand (A+C)^6 ?

The solution to expand (A+C)^6 is A^6+6A^5C+15A^4C^2+20A^3C^3+15A^2C^4+6AC^5+C^6

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expand (A+C)^6

expand

(A + C)^{6}

A^{6} + 6 A^{5} C + 15 A^{4} C^{2} + 20 A^{3} C^{3} + 15 A^{2} C^{4} + 6 A C^{5} + C^{6}

Solution steps

(A + C)^{6}

Apply binomial theorem:

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

a = A, b = C

= i = 0 \sum 6 (i 6) A^{(6 - i)} C^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} A^{6} C^{0} : A^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} A^{5} C^{1} : 6 A^{5} C

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} A^{4} C^{2} : 15 A^{4} C^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} A^{3} C^{3} : 20 A^{3} C^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} A^{2} C^{4} : 15 A^{2} C^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} A^{1} C^{5} : 6 A C^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} A^{0} C^{6} : C^{6}

= A^{6} + 6 A^{5} C + 15 A^{4} C^{2} + 20 A^{3} C^{3} + 15 A^{2} C^{4} + 6 A C^{5} + C^{6}

3 a b^{3} (a + b)

x^{3} + 9 x^{2} + 27 x + 19

(2 x + 3 y)^{7}

\frac{68}{5}

+ x^{3} - 2 x - 56

What is expand (A+C)^6 ?
The solution to expand (A+C)^6 is A^6+6A^5C+15A^4C^2+20A^3C^3+15A^2C^4+6AC^5+C^6