What is expand (m+n)^6 ?

The solution to expand (m+n)^6 is m^6+6m^5n+15m^4n^2+20m^3n^3+15m^2n^4+6mn^5+n^6

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expand (m+n)^6

expand

(m + n)^{6}

m^{6} + 6 m^{5} n + 15 m^{4} n^{2} + 20 m^{3} n^{3} + 15 m^{2} n^{4} + 6 m n^{5} + n^{6}

Solution steps

(m + n)^{6}

Apply binomial theorem:

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

a = m, b = n

= i = 0 \sum 6 (i 6) m^{(6 - i)} n^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} m^{6} n^{0} : m^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} m^{5} n^{1} : 6 m^{5} n

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} m^{4} n^{2} : 15 m^{4} n^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} m^{3} n^{3} : 20 m^{3} n^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} m^{2} n^{4} : 15 m^{2} n^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} m^{1} n^{5} : 6 m n^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} m^{0} n^{6} : n^{6}

= m^{6} + 6 m^{5} n + 15 m^{4} n^{2} + 20 m^{3} n^{3} + 15 m^{2} n^{4} + 6 m n^{5} + n^{6}

(3 - i) (4 + i)

\frac{1}{6} + \frac{1}{7}

\frac{4 p ^{- 8}}{- 4 ( p ^{- 2} ) ^{4}}

(2 x^{2} + 4 y)^{3}

(5 a + 2)^{2}

What is expand (m+n)^6 ?
The solution to expand (m+n)^6 is m^6+6m^5n+15m^4n^2+20m^3n^3+15m^2n^4+6mn^5+n^6