Binomial Theorem

Binomial Theorem: Level 4 Challenges


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

k=1504(4k1)(20154k1) \large\displaystyle \sum_{k=1}^{504} \left ( 4k-1 \right ) \binom{2015}{4k-1}

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

Evaluate p+q+r p+q+r .

Image Credit: Wikimedia TED-43.

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

Find the number of natural numbers which are not good.

Compute the remainder when (20141)+(20144)+(20147)++(20142014) \dbinom {2014}{1} + \dbinom {2014}{4} + \dbinom {2014}{7} + \cdots + \dbinom {2014}{2014} is divided by 1000 1000 .

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


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