Question 1158359
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            There is another way and another method to prove it,

            much more effective,  more easy and more straightforward  (and really nice (!) ).



<pre>
Use the binomial formula


    {{{(1 + x)^n}}} = 1 + {{{C[n]^1*x}}} + {{{C[n]^2*x^2}}} +{{{C[n]^3*x^3}}} + {{{C[n]^4*x^4}}} + . . . + {{{C[n]^(n-1)*x^(n-1)}}} + {{{x^n}}}.


Now plug in x= 1 into this formula.  You will get


    {{{2^n}}} = 1 + {{{C[n]^1}}} + {{{C[n]^2}}} +{{{C[n]^3}}} + {{{C[n]^4}}} + . . . + {{{C[n]^(n-1)}}} + 1,


and the proof is completed.



Those who saw this proof at least once in their life, will never forget it . . . 
</pre>

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See the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/Permutations/Remarkable-identities-for-Binomial-Coefficients.lesson>Remarkable identities for Binomial Coefficients</A>

in this site.


Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this textbook under the topic
"<U>Binomial expansion, binomial coefficients, Pascal's triangle</U>".



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.