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Binomial Expansion:
(n choose k) + (n choose (k+1))...need to use the factorial form to go through to step and eventually prove the answer... = ((n+1) choose (k+1))...
 one year ago
 one year ago
Binomial Expansion: (n choose k) + (n choose (k+1))...need to use the factorial form to go through to step and eventually prove the answer... = ((n+1) choose (k+1))...
 one year ago
 one year ago

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acrossBest ResponseYou've already chosen the best response.1
\[\begin{align} \binom nk+\binom n{k+1}&=\frac{n!}{k!(nk)!}+\frac{n!}{(k+1)!(nk1)!}\\ &=\frac{1}{nk}\left[\frac{n!(k+1)}{(k+1)!(nk1)!}+\frac{n!(nk)}{(k+1)!(nk1)!}\right]\\ &=\frac{(n+1)!}{(k+1)!(nk)!}=\binom{n+1}{k+1}. \end{align}\]
 one year ago

fpmakerBest ResponseYou've already chosen the best response.0
Thank you! My algebra with factorials is a little weak..is there any way you could briefly explain how you got to each step? Thanks again!!
 one year ago

acrossBest ResponseYou've already chosen the best response.1
The only algebraic manipulation that's worth mentioning is that \(n!=n(n1)!\). Both the LHS and the RHS on first line are selfexplanatory. On the second line, I factored out \((nk)^{1}\) from both terms (notice I multiplied the second term by \(nk\)) to equate denominators (using the idea I just mentioned). Finally, I just expanded and added the numerators on the third line to prove the equality.
 one year ago
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