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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 3

         

In the expansion of \((2x+\frac{k}{x})^8\), where \(k\) is a positive constant, the term independent of \(x\) is \(700000\). Find \(k.\)

\[\large \displaystyle \sum _{ r=0 }^{ n } { (-1 )}^{ r } { \binom{n}{r} }^{-1} \]

If \(n\) is an an odd positive integer, find the value of this sum.

\[\large \left ( \sqrt {71} +1 \right )^{71} - \left ( \sqrt {71} -1 \right )^{71} \]

What is the last digit of the number above?

What is the integral part of number \((\sqrt2+1)^6\)?

Details and Assumptions:

  • As an explicit example, the integral part of \(123.456\) is \(123\).

How many trailing zero(s) are there in the constant term of \( \left (x+\frac{1}{x} \right )^{2014}\)?

Details and Assumptions:

  • The number of trailing zeroes in a number is the number of zeroes at the end of the number, e.g., \(100\) has \(2\) trailing zeroes, and \(100000001\) has none.
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