What is expand (1+r)^6 ?

The solution to expand (1+r)^6 is 1+6r+15r^2+20r^3+15r^4+6r^5+r^6

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expand (1+r)^6

expand

(1 + r)^{6}

1 + 6 r + 15 r^{2} + 20 r^{3} + 15 r^{4} + 6 r^{5} + r^{6}

Solution steps

(1 + r)^{6}

Apply binomial theorem:

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

a = 1, b = r

= i = 0 \sum 6 (i 6) \cdot 1^{(6 - i)} r^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} \cdot 1^{6} r^{0} : 1

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} \cdot 1^{5} r^{1} : 6 r

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} \cdot 1^{4} r^{2} : 15 r^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} \cdot 1^{3} r^{3} : 20 r^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} \cdot 1^{2} r^{4} : 15 r^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} \cdot 1^{1} r^{5} : 6 r^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} \cdot 1^{0} r^{6} : r^{6}

= 1 + 6 r + 15 r^{2} + 20 r^{3} + 15 r^{4} + 6 r^{5} + r^{6}

(\frac{x ^{2} + 1}{x ^{2} + 2})^{4}

1 - (3 x^{15}) + 67

\frac{p ^{4}}{q} - \frac{q ^{2}}{p ^{2}}

(- 7 z)^{2}

a [13] + a [7] \cdot a [20] - a \frac{13}{100} a [2]

What is expand (1+r)^6 ?
The solution to expand (1+r)^6 is 1+6r+15r^2+20r^3+15r^4+6r^5+r^6