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fpmaker
 3 years 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))...
fpmaker
 3 years 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))...

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

fpmaker
 3 years ago
Best ResponseYou've already chosen the best response.0Thank 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!!

across
 3 years ago
Best ResponseYou've already chosen the best response.1The 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.
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