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Given that $\binom{17}{7}=19448$, $\binom{17}{8}=24310$ and $\binom{17}{9}=24310$, calculate $\binom{19}{9}$.

thanks!

 Apr 6, 2021
 #1
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Using Pascal's identity we have $\binom{17}{8}+\binom{17}{9}=\binom{18}{9}=24310+24310=48620$.

 

Now, to get from $\binom{18}{9}$ to $\binom{19}{9}$, you multiply by $\frac{19}{10}$. To see this, let's write out the binomial coefficients using factorials.

 

This gives you $\binom{18}{9} =\frac{18!}{9!*(18-9)!}=\frac{18!}{9!*9!}$.

 

Now, $\binom{19}{9}=\frac{19!=18!*19}{9!*(19-9)!=9!*9!*10}$. Hopefully you should see it by now.

 

So $\binom{19}{9}=\frac{48620*19}{10}=\boxed{92378}$.

 

However, there might be an easier way using more identities, so if anybody else has a solution that is easier, please speak up! :)

I think there might be more identites because my solution did not use the fact that $\binom{17}{7}=19448$

 Apr 6, 2021
 #2
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thank you so much!!! this was really helpful!

Guest Apr 6, 2021

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