Exam 11: Further Topics in Algebra

solve each problem involving counting theory. -A cheerleading squad consists of 6 boys and 5 girls. Six cheerleaders are to be selected to form a human pyramid. If the pyramid is made with 3 boys and 3 girls, how many ways can the six cheerleaders be selected?

solve each problem involving counting theory. -An experiment consists of rolling a die four times. Find the probability of each event. (a) Exactly 3 rolls result in a 6 . (b) All four rolls result in a 5.

solve each problem involving counting theory. -A rental car company offers 3 sizes of cars in 6 different models. How many different cars are there if each car comes with either manual or automatic transmission?

(a) Use the binomial theorem to expand $(3 x-y)^{4}$ . (b) Find the sixth term in the expansion of $(w-3 y)^{7}$ .

Find the sum of the first eight terms of the sequence described. (a) Arithmetic with $a_{1}=20$ and $d=-4$ . (b) Geometric with $a_{1}=3$ and $r=2$ .

In each sequence defined, find $a_{6}$ . (a) An arithmetic sequence with $a_{1}=7$ and $a_{3}=-1$ . (b) A geometric sequence with $a_{1}=5$ and $r=-\frac{1}{2}$ .

solve each problem involving counting theory. -A girl opens her tackle box while fishing to find that she has 6 different sizes of hooks, 5 sizes of lead sinkers, and 3 sizes of bobbers. How many different fishing set-ups can she make if she uses one of each?

solve each problem involving counting theory. -Two decks of standard playing cards, including 4 jokers, have a total of 108 cards. One card is drawn. (a) Find the probability of drawing a black face card (Jack, Queen, King). (b) Find the probability of drawing a joker or a red 7. (c) Find the probability of drawing a face card (Jack, Queen, King) or a 3. (d) What are the odds in favor of drawing the king of hearts?

solve each problem involving counting theory. -Suppose that a jeweler wishes to make rings which contain exactly 3 different birthstones in a line. Since there are 12 possible birthstones available, how many such rings are possible?

Write the first four terms for each sequence. State whether the sequence is arithmetic, geometric, or neither. (a) $a_{n}=(n+1)(n-2)$ (b) $a_{n}=\left(-\frac{1}{4}\right)^{n-1}$ (c) $a_{1}=2, a_{2}=6, a_{n}=2 a_{n-1}-a_{n-2}$ , for $n \geq 3$

In each sequence defined, find $a_{7}$ . (a) An arithmetic sequence with $a_{1}=5$ and $a_{3}=-3$ . (b) A geometric sequence with $a_{1}=-2$ and $r=\frac{1}{2}$ .

Use mathematical induction to prove that for all positive integers $n, 6+12+18+24+\cdots+6 n=3 n(n+1)$ .

solve each problem involving counting theory. -Suppose that a jeweler wishes to make rings which contain exactly 4 different birthstones in a line. Since there are 12 possible birthstones available, how many such rings are possible?

Evaluate each sum that exists. (a) $\sum_{i=1}^{25}(5-4 i)$ (b) $\sum_{i=1}^{6} \frac{3}{4}(2)^{i}$ (c) $\sum_{i=1}^{\infty} 2\left(\frac{3}{4}\right)^{i}$ (d) $\sum_{i=1}^{\infty}\left(\frac{4}{3}\right)^{i}$

Evaluate each sum that exists. (a) $\sum_{i=1}^{25}(20-3 i)$ (b) $\sum_{i=1}^{6} \frac{4}{3}(3)^{i}$ (c) $\sum_{i=1}^{\infty} 3\left(\frac{5}{4}\right)^{i}$ (d) $\sum_{i=1}^{\infty} 5\left(\frac{4}{5}\right)^{i}$

Write the first four terms for each sequence. State whether the sequence is arithmetic, geometric, or neither. (a) $a_{n}=(3 n-1)(n+1)$ (b) $a_{n}=\left(-\frac{3}{2}\right)^{n-1}$ (c) $a_{1}=1, a_{2}=6, a_{n}=2 a_{n-1}-a_{n-2}$ , for $n \geq 3$

solve each problem involving counting theory. -An experiment consists of rolling a die seven times. Find the probability of each event. (a) Exactly 2 rolls result in a 6. (b) None of the rolls results in a 6 .

Evaluate the following. (a) (b) $\left(\begin{array}{l}7 \\ 4\end{array}\right)$ (c) 8 ! (d) $P(5,4)$

(a) Use the binomial theorem to expand $(x-7 y)^{4}$ . (b) Find the fourth term in the expansion of $(w-3 y)^{9}$ .

solve each problem involving counting theory. -A high school soccer team must be made up of 7 seniors and 4 juniors. If 11 seniors and 9 juniors are eligible for the team, how many teams can be formed?