# How to solve:- binomial expansion of (1+2x)^7

Welcome to my article binomial expansion of (1+2x)^7. This question is taken from the simplification lesson.
The solution of this question has been explained in a very simple way by a well-known teacher by doing addition, subtraction, and fractions.
For complete information on how to solve this question binomial expansion of (1+2x)^7, read and understand it carefully till the end.

Let us know how to solve this question binomial expansion of (1+2x)^7.

First write the question on the page of the notebook.

## binomial expansion of (1+2x)^7

write out the expansoions of the following \displaystyle {{\left( {1+2x} \right)}^{7}}

given dinomail formula

or,

## \displaystyle {{\left( {a+b} \right)}^{n}}={{a}^{n}}+n{{a}^{{n-1}}}{{b}^{1}}+\frac{{n\left( {n-1} \right){{a}^{{n-2}}}{{b}^{2}}}}{{2!}}+\frac{{n\left( {n-1} \right)\left( {n-2} \right){{a}^{{n-3}}}{{b}^{3}}}}{{3!}}+\frac{{n\left( {n-1} \right)\left( {n-2} \right)\left( {n-3} \right){{a}^{{n-4}}}{{b}^{4}}}}{{4!}}+\frac{{n\left( {n-1} \right)\left( {n-2} \right)\left( {n-3} \right)\left( {n-4} \right){{a}^{{n-5}}}{{b}^{5}}}}{{5!}}+\frac{{n\left( {n-1} \right)\left( {n-2} \right)\left( {n-3} \right)\left( {n-4} \right)\left( {n-5} \right){{a}^{{n-6}}}{{b}^{6}}}}{{6!}}+\frac{{n\left( {n-1} \right)\left( {n-2} \right)\left( {n-3} \right)\left( {n-4} \right)\left( {n-5} \right)\left( {n-6} \right){{a}^{{n-7}}}{{b}^{7}}}}{{7!}}

here,

a=1

b=2x

n=7

then,

\displaystyle {{\left( {1+2x} \right)}^{7}}={{1}^{7}}+7{{(1)}^{{7-1}}}{{(2x)}^{1}}+\frac{{7\left( {7-1} \right){{{(1)}}^{{7-2}}}{{{(2x)}}^{2}}}}{{2!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right){{{(1)}}^{{7-3}}}{{{(2x)}}^{3}}}}{{3!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right){{{(1)}}^{{7-4}}}{{{(2x)}}^{4}}}}{{4!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right){{{(1)}}^{{7-5}}}{{{(2x)}}^{5}}}}{{5!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right)\left( {7-5} \right){{{(1)}}^{{7-6}}}{{{(2x)}}^{6}}}}{{6!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right)\left( {7-5} \right)\left( {7-6} \right){{{(1)}}^{{7-7}}}{{{(2x)}}^{7}}}}{{7!}}

\displaystyle {{\left( {1+2x} \right)}^{7}}={{1}^{7}}+7{{(1)}^{6}}{{(2x)}^{1}}+\frac{{7\left( {7-1} \right){{{(1)}}^{5}}{{{(2x)}}^{2}}}}{{2!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right){{{(1)}}^{4}}{{{(2x)}}^{3}}}}{{3!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right){{{(1)}}^{3}}{{{(2x)}}^{4}}}}{{4!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right){{{(1)}}^{2}}{{{(2x)}}^{5}}}}{{5!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right)\left( {7-5} \right){{{(1)}}^{1}}{{{(2x)}}^{6}}}}{{6!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right)\left( {7-5} \right)\left( {7-6} \right){{{(1)}}^{0}}{{{(2x)}}^{7}}}}{{7!}}

\displaystyle {{\left( {1+2x} \right)}^{7}}=1+7(1){{(2x)}^{1}}+\frac{{7\left( {7-1} \right)(1){{{(2x)}}^{2}}}}{{2!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)(1){{{(2x)}}^{3}}}}{{3!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)(1){{{(2x)}}^{4}}}}{{4!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right)(1){{{(2x)}}^{5}}}}{{5!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right)\left( {7-5} \right)(1){{{(2x)}}^{6}}}}{{6!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right)\left( {7-5} \right)\left( {7-6} \right)(1){{{(2x)}}^{7}}}}{{7!}}
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\displaystyle {{\left( {1+2x} \right)}^{7}}=1+7{{(2x)}^{1}}+\frac{{7\left( {7-1} \right){{{(2x)}}^{2}}}}{{2!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right){{{(2x)}}^{3}}}}{{3!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right){{{(2x)}}^{4}}}}{{4!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right){{{(2x)}}^{5}}}}{{5!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right)\left( {7-5} \right){{{(2x)}}^{6}}}}{{6!}}+\frac{{7\left( {7-1} \right)\left( {7-2} \right)\left( {7-3} \right)\left( {7-4} \right)\left( {7-5} \right)\left( {7-6} \right){{{(2x)}}^{7}}}}{{7!}}

\displaystyle {{\left( {1+2x} \right)}^{7}}=1+7{{(2x)}^{1}}+\frac{{7\left( 6 \right){{{(2x)}}^{2}}}}{{2!}}+\frac{{7\left( 6 \right)\left( 5 \right){{{(2x)}}^{3}}}}{{3!}}+\frac{{7\left( 6 \right)\left( 5 \right)\left( 4 \right){{{(2x)}}^{4}}}}{{4!}}+\frac{{7\left( 6 \right)\left( 5 \right)\left( 4 \right)\left( 3 \right){{{(2x)}}^{5}}}}{{5!}}+\frac{{7\left( 6 \right)\left( 5 \right)\left( 4 \right)\left( 3 \right)\left( 2 \right){{{(2x)}}^{6}}}}{{6!}}+\frac{{7\left( 6 \right)\left( 5 \right)\left( 4 \right)\left( 3 \right)\left( 2 \right)\left( 1 \right){{{(2x)}}^{7}}}}{{7!}}

\displaystyle {{\left( {1+2x} \right)}^{7}}=1+7{{(2x)}^{1}}+\frac{{42{{{(2x)}}^{2}}}}{{2!}}+\frac{{42\left( 5 \right){{{(2x)}}^{3}}}}{{3!}}+\frac{{42\left( 5 \right)\left( 4 \right){{{(2x)}}^{4}}}}{{4!}}+\frac{{42\left( 5 \right)\left( 4 \right)\left( 3 \right){{{(2x)}}^{5}}}}{{5!}}+\frac{{42\left( 5 \right)\left( 4 \right)\left( 3 \right)\left( 2 \right){{{(2x)}}^{6}}}}{{6!}}+\frac{{42\left( 5 \right)\left( 4 \right)\left( 3 \right)\left( 2 \right)\left( 1 \right){{{(2x)}}^{7}}}}{{7!}}

\displaystyle {{\left( {1+2x} \right)}^{7}}=1+7{{(2x)}^{1}}+\frac{{42{{{(2x)}}^{2}}}}{{2!}}+\frac{{42\left( 5 \right){{{(2x)}}^{3}}}}{{3!}}+\frac{{42\left( {20} \right){{{(2x)}}^{4}}}}{{4!}}+\frac{{42\left( {20} \right)\left( 3 \right){{{(2x)}}^{5}}}}{{5!}}+\frac{{42\left( {20} \right)\left( 3 \right)\left( 2 \right){{{(2x)}}^{6}}}}{{6!}}+\frac{{42\left( {20} \right)\left( 3 \right)\left( 2 \right)\left( 1 \right){{{(2x)}}^{7}}}}{{7!}}

\displaystyle {{\left( {1+2x} \right)}^{7}}=1+7{{(2x)}^{1}}+21{{(2x)}^{2}}+14\left( 5 \right){{(2x)}^{3}}+42\left( 5 \right){{(2x)}^{4}}+42\left( 4 \right)\left( 3 \right){{(2x)}^{5}}+7\left( {20} \right)\left( 3 \right)\left( 2 \right){{(2x)}^{6}}+6\left( {20} \right)\left( 3 \right)\left( 2 \right)\left( 1 \right){{(2x)}^{7}}

\displaystyle {{\left( {1+2x} \right)}^{7}}=1+7{{(2x)}^{1}}+21{{(2x)}^{2}}+70{{(2x)}^{3}}+210{{(2x)}^{4}}+42\left( {12} \right){{(2x)}^{5}}+7\left( {20} \right)\left( 6 \right){{(2x)}^{6}}+6\left( {20} \right)\left( 6 \right){{(2x)}^{7}}

\displaystyle {{\left( {1+2x} \right)}^{7}}=1+7{{(2x)}^{1}}+21{{(2x)}^{2}}+70{{(2x)}^{3}}+210{{(2x)}^{4}}+42\left( {12} \right){{(2x)}^{5}}+\left( {20} \right)\left( {42} \right){{(2x)}^{6}}+\left( {20} \right)\left( {36} \right){{(2x)}^{7}}

\displaystyle {{\left( {1+2x} \right)}^{7}}=1+7{{(2x)}^{1}}+21{{(2x)}^{2}}+70{{(2x)}^{3}}+210{{(2x)}^{4}}+504{{(2x)}^{5}}+84{{(2x)}^{6}}+720{{(2x)}^{7}}

\displaystyle {{\left( {1+2x} \right)}^{7}}=1+7(2x)+21(4{{x}^{2}})+70(8{{x}^{3}})+210(16{{x}^{4}})+504(32{{x}^{5}})+84(64{{x}^{6}})+720(128{{x}^{7}})