91Ó°ÊÓ

Lottery. In a state lottery in which 6 numbers are drawn from a possible 40 numbers, the number of possible 6 -number combinations is equal to \(\left(\begin{array}{c}40 \\ 6\end{array}\right) .\) How many possible combinations are there?

Write the first four terms of the sequence defined by each recursion formula. Assume the sequence begins at \(n=1\). $$a_{1}=20 \quad a_{n}=\frac{a_{n-1}}{n^{2}}$$

Lottery. In a state lottery in which 6 numbers are drawn from a possible 60 numbers, the number of possible 6 -number combinations is equal to \(\left(\begin{array}{c}60 \\ 6\end{array}\right) .\) How many possible combinations are there?

Poker. With a deck of 52 cards, 5 cards are dealt in a game of poker. There are a total of \(\left(\begin{array}{c}52 \\ 5\end{array}\right)\) different 5 -card poker hands that can be dealt. How many possible hands are there?

Find the sum of each infinite geometric series, if possible. $$\sum_{n=0}^{\infty}-9\left(\frac{1}{3}\right)^{n}$$

Write the first four terms of the sequence defined by each recursion formula. Assume the sequence begins at \(n=1\). $$a_{1}=1, a_{2}=2 \quad a_{n}=a_{n-1} \cdot a_{n-2}$$

Find the sum of each infinite geometric series, if possible. $$\sum_{n=0}^{\infty}-8\left(-\frac{1}{2}\right)^{n}$$

Write the first four terms of the sequence defined by each recursion formula. Assume the sequence begins at \(n=1\). $$a_{1}=1, a_{2}=2 \quad a_{n}=\frac{a_{n-2}}{a_{n-1}}$$

Explain the mistake that is made. Evaluate the expression \(\left(\begin{array}{l}7 \\ 5\end{array}\right)\) Solution: Write out the \(\begin{array}{l}\text { binomial coefficient } \\ \text { in terms of factorials. }\end{array}\left(\begin{array}{l}7 \\\ 5\end{array}\right)=\frac{7 !}{5 !}\) \(\begin{array}{l}\text { Write out the } \\ \text { factorials. }\end{array}\left(\begin{array}{l}7 \\ 5\end{array}\right)=\frac{7 !}{5 !}=\frac{7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}\) Simplify. \(\left(\begin{array}{l}7 \\ 5\end{array}\right)=\frac{7 !}{5 !}=\frac{7 \cdot 6}{1}=42\) This is incorrect. What mistake was made?

Write the first four terms of the sequence defined by each recursion formula. Assume the sequence begins at \(n=1\). $$a_{1}=1, a_{2}=-1 \quad a_{n}=(-1)^{n}\left[a_{n-1}^{2}+a_{n-2}^{2}\right]$$