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# Binomial Theorem

If a coin comes up heads you win $10, but if it comes up tails you win$0. How likely are you to win exactly \$30 in five flips?

# Binomial Theorem: Level 4 Challenges

$\large \sum_{k=0}^n \left[ (-1)^k \dbinom{n}{k} (n-k)^n \right] = \ ?$

$\large\displaystyle \sum_{k=1}^{504} \left ( 4k-1 \right ) \binom{2015}{4k-1}$

If the sum above can be written as $$p.q^{r}$$, where $$p$$, $$q$$ and $$r$$ are positive integers with $$q$$ being a prime.

Evaluate $$p+q+r$$.

###### Image Credit: Wikimedia TED-43.

Let a natural number $$n$$ be good if there exist two distinct non-integral real numbers $$a$$ and $$b$$ such that $$a^k - b^k$$ is an integer for all $$1 \leq k \leq n$$.

Find the number of natural numbers which are not good.

Compute the remainder when $\dbinom {2014}{1} + \dbinom {2014}{4} + \dbinom {2014}{7} + \cdots + \dbinom {2014}{2014}$ is divided by $$1000$$.

Find the number of odd coefficients in the expansion of $$(a + b)^{2015}.$$

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