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Question-190857




Question Number 190857 by AlbertEinstein999 last updated on 13/Apr/23
Commented by mr W last updated on 13/Apr/23
Σ_(k=0) ^n C_n ^k =2^n −1  Σ_(k=0) ^n (C_n ^k )^2 =C_(2n) ^n
$$\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{C}_{{n}} ^{{k}} =\mathrm{2}^{{n}} \cancel{−\mathrm{1}} \\ $$$$\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left({C}_{{n}} ^{{k}} \right)^{\mathrm{2}} ={C}_{\mathrm{2}{n}} ^{{n}} \\ $$
Commented by mehdee42 last updated on 13/Apr/23
why  2^n −1 ?    Σ_(k=0) ^n C_n ^k =2^n
$${why}\:\:\mathrm{2}^{{n}} −\mathrm{1}\:?\:\: \\ $$$$\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{C}_{{n}} ^{{k}} =\mathrm{2}^{{n}} \\ $$$$ \\ $$
Commented by mr W last updated on 13/Apr/23
yes.  i had Σ_(k=1) ^n C_n ^k  in head.
$${yes}. \\ $$$${i}\:{had}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{C}_{{n}} ^{{k}} \:{in}\:{head}. \\ $$

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