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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?

Properties of Binomial Coefficients

         

What positive integer \(n\) satisfies \[{{n+2} \choose 11}=109 \cdot {n \choose 9}?\]

Caroline has \(417\) coins, no two of which are the same. The number of ways she can choose \(89\) of these coins is the same as the number of ways she can choose \(k\) of these coins for some \(k > 89.\) What is the value of \(k?\)

What is the arithmetic mean of the following sereis: \[{ 8 \choose 0 }, { 8 \choose 1 }, { 8 \choose 2 }, \cdots, { 8 \choose 8 }?\]

What is the value of positive integer \(n\) satisfying \[{26 \choose n}={26 \choose {n-4}}?\]

If \({n+4 \choose 2}+{n+3 \choose 2}=25,\) what is the value of \(n?\)

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