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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?

Level 4

         

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

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

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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