Chapter 12 - Sequences; Induction; the Binomial Theorem - 12.5 The Binomial Theorem - 12.5 Assess Your Understanding - Page 837: 47

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We have to prove the identity: $\binom{n}{0}+\binom{n}{1}+...+\binom{n}{n}=2^n$ We have: $(1+1)^n=2^n$ Use the Binomial Theorem to expand the left side: $\binom{n}{0}1^n1^0+\binom{n}{1}1^{n-1}1^1+\binom{n}{2}1^{n-2}1^2+.....+\binom{n}{n}1^01^n=2^n$ $\binom{n}{0}+\binom{n}{1}+...+\binom{n}{n}=2^n$ We proved the given identity.