Algebra
# Binomial Theorem

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

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

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 \( p+q+r \).

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**.

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