Chapter 10

Wardrobe mix ' \(n\) ' match A girl has four skirts and six blouses. How many different skirt-blouse combinations can she wear?

Exer. 19-22: Find the sum \(S_{n}\) of the arithmetic sequence that satisfies the stated conditions. $$ a_{1}=40, \quad d=-3, \quad n=30 $$

Given a geometric sequence with \(a_{2}=3\) and \(a_{5}=-81\), find \(r\) and \(a_{9}\).

In how many different ways can seven keys be arranged on a key ring if the keys can slide completely around the ring?

Use the binomial theorem to expand and simplify. $$ (x+y)^{4} $$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). 2 is a factor of \(n^{2}+n\).

Exer. 19-22: Find the sum \(S_{n}\) of the arithmetic sequence that satisfies the stated conditions. $$ a_{1}=5, \quad d=0.1, \quad n=40 $$

Find the sum. $$\sum_{k=1}^{10} 3^{2}$$

A committee of 3 men and 2 women is to be chosen from a group of 12 men and 8 women. Determine the number of different ways of selecting the committee.

Use the binomial theorem to expand and simplify. $$ (x-y)^{7} $$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). 4 is a factor of \(5^{n}-1\).

License plate numbers In a certain state, automobile license plates start with one letter of the alphabet, followed by five digits \((0,1,2, \ldots, 9)\). Find how many different license plates are possible if (a) the first digit following the letter cannot be 0 (b) the first letter cannot be \(\mathrm{O}\) or \(\mathrm{I}\) and the first digit cannot be 0

Exer. 19-22: Find the sum \(S_{n}\) of the arithmetic sequence that satisfies the stated conditions. $$ a_{1}=-9, \quad a_{10}=15, \quad n=10 $$

\(a_{1}=2, \quad a_{k+1}=3 a_{k}-5\)

Find the sum. $$\sum_{k=1}^{9}(-\sqrt{5})^{k}$$

Let the letters \(\mathrm{G}\) and \(\mathrm{B}\) denote a girl birth and a boy birth, respectively. For a family of three boys and three girls, one possible birth order is \(\mathrm{G} \mathrm{G} \mathrm{G} \mathrm{B} \mathrm{B} \mathrm{B.} \mathrm{How} \mathrm{many} \mathrm{}\) birth orders are possible for these six children?

Use the binomial theorem to expand and simplify. (x-y)^{5}

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). 9 is a factor of \(10^{n+1}+3 \cdot 10^{n}+5 .\)

Tossing dice Two dice are tossed, one after the other. In how many different ways can they fall? List the number of different ways the sum of the dots can equal (a) 3 (b) 5 (c) 7 (d) 9 (e) 11

Exer. 19-22: Find the sum \(S_{n}\) of the arithmetic sequence that satisfies the stated conditions. $$ a_{7}=\frac{7}{3}, \quad d=-\frac{2}{3}, \quad n=15 $$

\(a_{1}=5, \quad a_{k+1}=7-2 a_{k}\)

Find the sum. $$\sum_{k=0}^{9}\left(-\frac{1}{2}\right)^{k+1}$$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). If \(a\) is greater than 1 , then \(a^{n}>1\).

Seating arrangement A row of six seats in a classroom is to be filled by selecting individuals from a group of ten students. (a) In how many different ways can the seats be occupied? (b) If there are six boys and four girls in the group and if boys and girls are to be alternated, find the number of different seating arrangements.

Exer. 23-28: Find the sum. $$ \sum_{k=1}^{20}(3 k-5) $$

\(a_{1}=-3, \quad a_{k+1}=a_{k}^{2}\)

Find the sum. $$\sum_{k=1}^{7}\left(3^{-k}\right)$$

Use the binomial theorem to expand and simplify. $$ (2 t-s)^{5} $$

Scheduling courses A student in a certain college may take mathematics at \(8,10,11\), or 2 o'clock; English at \(9,10,1\), or 2 ; and history at \(8,11,2\), or 3 . Find the number of different ways in which the student can schedule the three courses.

Exer. 23-28: Find the sum. $$ \sum_{k=1}^{12}(7-4 k) $$

\(a_{1}=128, \quad a_{k+1}=\frac{1}{4} a_{k}\)

If a single die is tossed, find the probability of obtaining an odd number or a prime number.

To win a state lottery game, a player must correctly select six numbers from the numbers 1 through 49 . (a) Find the total number of selections possible. (b) Work part (a) if a player selects only even numbers.

Use the binomial theorem to expand and simplify. $$ \left(\frac{1}{3} x+y^{2}\right)^{5} $$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). \(a-b\) is a factor of \(a^{n}-b^{n}\). (Hint: \(\left.a^{k+1}-b^{k+1}=a^{k}(a-b)+\left(a^{k}-b^{k}\right) b .\right)\)

True-or-false test In how many different ways can a test consisting of ten true-or-false questions be completed?

Exer. 23-28: Find the sum. $$ \sum_{k=1}^{18}\left(\frac{1}{2} k+7\right) $$

A mathematics department has ten faculty members but only nine offices, so one office must be shared by two individuals. In how many different ways can the offices be assigned?

Use the binomial theorem to expand and simplify. $$ 5\left(\frac{1}{2} x+y^{3}\right)^{4} $$

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ a+b \text { is a factor of } a^{2 n-1}+b^{2 n-1} $$

Multiple-choice test A test consists of six multiple-choice questions, and there are five choices for each question. In how many different ways can the test be completed?

Exer. 23-28: Find the sum. $$ \sum_{k=1}^{10}\left(\frac{1}{4} k+3\right) $$

\( a_{1}=3, \quad a_{k+1}=1 / a_{k}\)

Express the sum in terms of summation notation. (Answers are not unique.) $$2+4+8+16+32+64+128$$

If the probability of a baseball player's getting a hit in one time at bat is \(0.326\), find the probability that the player gets no hits in 4 times at bat.

In a round-robin tennis tournament, every player meets every other player exactly once. How many players can participate in a tournament of 45 matches?

Use the binomial theorem to expand and simplify. $$ \left(\frac{1}{x^{2}}+3 x\right)^{6} $$

Exer. 27-32: Find the smallest positive integer \(j\) for which the statement is true. Use the extended principle of mathematical induction to prove that the formula is true for every integer greater than \(j\). $$ n+12 \leq n^{2} $$

Seating arrangement In how many different ways can eight people be seated in a row?

\(a_{1}=2, \quad a_{k+1}=\left(a_{k}\right)^{k}\) $$