Chapter 6

Let \(U=|\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}|\) be the sample space of an experiment, where the outcomes are equally likely. Find the probability of each event. $$\\{a, b\\}$$

The \(n\) th Fibonacci number \(F_{n}\) is given by the sum of the numbers along the \(n\) th northeast diagonal of Pascal's triangle; that is, $$F_{n}=\sum_{i=0}^{\lfloor(n-1) / 2\rfloor}\left(\begin{array}{c} n-i-1 \\ i \end{array}\right)$$ Using this formula, compute each Fibonacci number. $$F_{6}$$

There are 15 rabbits in a cage. Five of them are injected with a certain drug. Three of the 15 rabbits are selected successively at random for an experiment. Find the probability that: Only the first rabbit is injected with the drug.

The Bell numbers \(B_{n},\) named after the English mathematician Eric T. Bell (1883-1960) and used in combinatorics, are defined recursively as follows: $$B_{0}=1$$ $$B_{n}=\sum_{i=0}^{n-1}\left(\begin{array}{c} n-1 \\ i \end{array}\right) B_{i}, \quad n \geq 1$$ Compute each Bell number. $$B_{2}$$

Let \(U=\\{a, b, c, d, e\\}\) be the sample space of an experiment, where the outcomes are equally likely. Find the probability of each event. $$\\{\mathbf{a}, \mathbf{c}, \mathrm{d}\\}$$

Find the number of solutions to each equation, where the variables are nonnegative integers. $$x_{1}+x_{2}+x_{3}+x_{4}=10$$

The password for a computer system consists of eight distinct alphabetic characters. Find the number of passwords possible that: Begin with the string CREAM.

An old zip code in the United States consists of five digits. Find the total number of possible zip codes that: Are palindromes.

Let \(U=|\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}|\) be the sample space of an experiment, where the outcomes are equally likely. Find the probability of each event. $$\\{\mathrm{a}, \mathrm{c}, \mathrm{d}\\}$$

Find the number of ways each sum can be formed from a collection of 10 nickels and 5 quarters. 25 cents

There are 15 rabbits in a cage. Five of them are injected with a certain drug. Three of the 15 rabbits are selected successively at random for an experiment. Find the probability that: Only the second rabbit is injected with the drug.

The Bell numbers \(B_{n},\) named after the English mathematician Eric T. Bell (1883-1960) and used in combinatorics, are defined recursively as follows: $$B_{0}=1$$ $$B_{n}=\sum_{i=0}^{n-1}\left(\begin{array}{c} n-1 \\ i \end{array}\right) B_{i}, \quad n \geq 1$$ Compute each Bell number. $$B_{3}$$

Find the number of solutions to each equation, where the variables are nonnegative integers. $$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=11$$

Find the number of ways each sum can be formed from a collection of 10 nickels and 5 quarters. 30 cents

A zip code in Canada consists of three letters and three digits. Each zip code begins with a letter. The letters and digits alternate; for instance, A1B2C3. Find the number of zip codes that: End in 6.

The password for a computer system consists of eight distinct alphabetic characters. Find the number of passwords possible that: Contain the word COMPUTER as a substring.

Let \(U=|\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}|\) be the sample space of an experiment, where the outcomes are equally likely. Find the probability of each event. $$\emptyset$$

There are 15 rabbits in a cage. Five of them are injected with a certain drug. Three of the 15 rabbits are selected successively at random for an experiment. Find the probability that: Only the first two rabbits are injected with the drug.

The Bell numbers \(B_{n},\) named after the English mathematician Eric T. Bell (1883-1960) and used in combinatorics, are defined recursively as follows: $$B_{0}=1$$ $$B_{n}=\sum_{i=0}^{n-1}\left(\begin{array}{c} n-1 \\ i \end{array}\right) B_{i}, \quad n \geq 1$$ Compute each Bell number. $$B_{4}$$

Find the number of solutions to each equation, where \(x_{i} \geq 1.\) $$x_{1}+x_{2}+x_{3}+x_{4}=11$$

Jane has two nickels, four dimes, three quarters, and two half-dollars in her handbag. Find the number of ways she can tip the waiter if she would like to give him: Exactly three coins.

A zip code in Canada consists of three letters and three digits. Each zip code begins with a letter. The letters and digits alternate; for instance, A1B2C3. Find the number of zip codes that: Begin with A and end in 3.

A survey of 475 customers at Chestnut Restaurant shows that of the three ice cream flavors - chocolate, strawberry, and vanilla - 65 like only chocolate, 75 like only strawberry, 85 like only vanilla, 100 like chocolate but not strawberry, 120 like strawberry but not vanilla, 140 like vanilla but not chocolate, and 65 like none of the flavors. A customer is selected at random from the survey. Find the probability that he likes: All flavors.

There are 15 rabbits in a cage. Five of them are injected with a certain drug. Three of the 15 rabbits are selected successively at random for an experiment. Find the probability that: Only the last two rabbits are injected with the drug.

The Bell numbers \(B_{n},\) named after the English mathematician Eric T. Bell (1883-1960) and used in combinatorics, are defined recursively as follows: $$B_{0}=1$$ $$B_{n}=\sum_{i=0}^{n-1}\left(\begin{array}{c} n-1 \\ i \end{array}\right) B_{i}, \quad n \geq 1$$ Compute each Bell number. $$B_{5}$$

A survey of 475 customers at Chestnut Restaurant shows that of the three ice cream flavors - chocolate, strawberry, and vanilla - 65 like only chocolate, 75 like only strawberry, 85 like only vanilla, 100 like chocolate but not strawberry, 120 like strawberry but not vanilla, 140 like vanilla but not chocolate, and 65 like none of the flavors. A customer is selected at random from the survey. Find the probability that he likes: Chocolate.

Find the number of solutions to each equation, where \(x_{i} \geq 1.\) $$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=13$$

Using the binomial theorem, prove each. $$ 2^{4 n}+3 n-1 \text { is divisible by } 9 . \text { Hint: } 2=3-1 . $$

A survey of 475 customers at Chestnut Restaurant shows that of the three ice cream flavors - chocolate, strawberry, and vanilla - 65 like only chocolate, 75 like only strawberry, 85 like only vanilla, 100 like chocolate but not strawberry, 120 like strawberry but not vanilla, 140 like vanilla but not chocolate, and 65 like none of the flavors. A customer is selected at random from the survey. Find the probability that he likes: Exactly two flavors.

Jane has two nickels, four dimes, three quarters, and two half-dollars in her handbag. Find the number of ways she can tip the waiter if she would like to give him: Not more than three coins.

Using the recursive definition of \(c_{n},\) compute each. $$\mathcal{C}_{4}$$

A zip code in Canada consists of three letters and three digits. Each zip code begins with a letter. The letters and digits alternate; for instance, A1B2C3. Find the number of zip codes that: Are possible.

Using the binomial theorem, prove each. \(2^{4 n}+3 n-1\) is divisible by \(9 .\) (Hint: \(2=3-1 .\))

The password for a computer system consists of eight distinct alphabetic characters. Find the number of passwords possible that: Contain the strings BLACK and WHITE.

Using the binomial theorem, prove each. \(4^{2 n}+10 n-1\) is divisible by \(25 .\) (Hint: \(4=5-1 .\))

A survey of 475 customers at Chestnut Restaurant shows that of the three ice cream flavors - chocolate, strawberry, and vanilla - 65 like only chocolate, 75 like only strawberry, 85 like only vanilla, 100 like chocolate but not strawberry, 120 like strawberry but not vanilla, 140 like vanilla but not chocolate, and 65 like none of the flavors. A customer is selected at random from the survey. Find the probability that he likes: Exactly one flavor.

Jane has two nickels, four dimes, three quarters, and two half-dollars in her handbag. Find the number of ways she can tip the waiter if she would like to give him: Exactly 50 cents.

Using the recursive definition of \(c_{n},\) compute each. $$\mathcal{C} 5$$

The password for a computer system consists of six alphanumeric characters and begins with a letter. Find the total number of passwords that: Are possible.

The password for a computer system consists of eight distinct alphabetic characters. Find the number of passwords possible that: Do not contain the string SAMPLE.

Find the number of solutions to each equation. $$x_{1}+x_{2}+x_{3}=10, x_{1} \geq 3,1 \leq x_{2} \leq 3, x_{3} \geq 5$$

Two dice are rolled. If a sum of six appears, Randy gets \(\$ 6\) from Wanda; otherwise, he loses \(\$ 3\) to her. Compute Randy's expected winnings.

Using the binomial theorem, prove each. \(\sum_{r=0}^{n}\left(\begin{array}{c}2 n \\ 2 r\end{array}\right)=\sum_{r=1}^{n}\left(\begin{array}{c}2 n \\ 2 r-1\end{array}\right)(\text {Hint: Use Theorem } 6.18.)\)

Jane has two nickels, four dimes, three quarters, and two half-dollars in her handbag. Find the number of ways she can tip the waiter if she would like to give him: Not more than 50 cents, using only one type of coin.

Using the recursive definition of \(c_{n},\) compute each. $$c_{6}$$

A botanist would like to plant three coleus, four zinnias, and five dahlias in a row in her front garden. How many ways can she plant them if: They can be planted in any order.

A zip code in Canada consists of three letters and three digits. Each zip code begins with a letter. The letters and digits alternate; for instance, A1B2C3. Find the number of zip codes that: End in RED.

Find the number of solutions to each equation. $$x_{1}+x_{2}+x_{3}=12, x_{1}, x_{2} \geq 5,1 \leq x_{3} \leq 4$$

Using the binomial theorem, prove each. \(\sum_{r=0}^{n} 2^{r}\left(\begin{array}{l}n \\ r\end{array}\right)=3^{n}\)

Using the recursive definition of \(c_{n},\) compute each. $$c_{8}$$