how do I figure out Which of the following is true?

Question

I am studying for my exam and I am kind of stuck on this question, how is it that the answer is a)? can someone explain this please.

1. Which one of the following is true?

   a) $$\sum_{k=0}^{n} {5^k}{n\choose k}= 6^n$$ b) $$\sum_{k=0}^{n} {4^{n-k}}5^k{n\choose k}= 8^n$$ c) $$\sum_{k=0}^{n} {5^k}{n\choose k}= 5^n$$ d) $$\sum_{k=0}^{n} {4^{k}}5^{n-k}{n\choose k}= 20^n$$

Answer 1

hint: Try to co-relate with Binomial theorem $$(x+y)^n=\sum_{k=0}^{n}\binom{n}{k}x^ky^{n-k}$$where $n$ is a positive integer. Notice that $\binom{n}{k}5^k$ when written as $\binom{n}{k}5^k1^{n-k}$ reveals more structure.

Answer 2

you can use the Binomial Theorem to answer the question. Check it out here: http://en.wikipedia.org/wiki/Binomial_theorem

greetings,

calculus