# Tagged Questions

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### Evaluate the sum $\sum_{0\leq j < k\leq n}\binom{n}{j}\binom{n}{k}$

Could someone give me a hint on how to do this? I believe I know what the answer to be (I computed some low values and checked on OEIS). However, I was hoping someone would be able to explain to me ...
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The chance of a rose flower blooming is .28. You are going to plant 5 rose flowers, what are the chances of 4 of them blooming? I was thinking the answer would be 35% since 28%x5=140 and 140/4=35. ...
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### Binomial expansion, the greatest term…

My question is related to binomial expansion, and more precisely the greatest term in expansion. Is it right that the formula for finding the greatest term is $$T_k\ge T_{k+1}$$? Now going to the ...
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### Binomial Summation

The sum $$1 + {n \choose 1}\cos \theta + {n \choose 2}\cos 2\theta + \cdots+ {n \choose n}\cos n\theta$$ is? I try to write this as the real part of $(1 + \cos \theta + i\sin \theta)^n$ but then ...
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### Find the 100th derivative of $x \sinh(2x)$
If $f(x) = x \sinh(2x)$, find $f^{({100})}(x)$. My (Incorrect) working so far: Using Leibniz' Formula for derivatives: $$(fg)^{(n)}=\sum_{k=0}^n{n\choose k}f^{(k)}g^{(n-k)}$$ ...