Chapter 6

Mark each sentence as true or false, where \(n\) is an arbitrary nonnegative integer and \(0 \leq r \leq n\). $$P(n, r)=P(n, n-r)$$

Two cards are drawn at random from a standard deck of cards. Find the probability that: Both are clubs.

Suppose that on the first day of Christmas you sent your love 1 gift, \(1+2\) gifts on the second day, \(1+2+3\) gifts on the third day, and so on. Show that the number of gifts sent on the \(n\) th day is \(C(n+1,2),\) where \(1 \leq n \leq 12\).

Two cards are drawn at random successively from a standard deck. The first card is replaced before the second is drawn. Find the probability that: Both are queens.

Two cards are drawn at random from a standard deck of cards. Find the probability that: One is a king and the other a queen.

Find the largest binomial coefficient in the expansion of each. $$(x+y)^{8}$$

(Twelve Days of Christmas) Suppose that on the first day of Christmas you sent your love 1 gift, \(1+2\) gifts on the second day, \(1+2+3\) gifts on the third day, and so on. Show that the total number of gifts sent by the \(n\) th day is \(C(n+2,3)\) , where \(1 \leq n \leq 12\) .

A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that: Begin with a vowel.

Mark each sentence as true or false, where \(n\) is an arbitrary nonnegative integer and \(0 \leq r \leq n\). \(n !\) is divisible by 10 if \(n>4\)

Suppose that on the first day of Christmas you sent your love 1 gift, \(1+2\) gifts on the second day, \(1+2+3\) gifts on the third day, and so on. Show that the total number of gifts sent by the \(n\) th day is \(C(n+2,3)\) where \(1 \leq n \leq 12\).

Two cards are drawn at random successively from a standard deck. The first card is replaced before the second is drawn. Find the probability that: Both are clubs.

Two cards are drawn at random from a standard deck of cards. Find the probability that: One is a club and the other a diamond.

Using Exercises 13-16, predict the largest binomial coefficient in the expansion of \((x+y)^{n}.\)

Solve each equation, where \(n \geq 0\). $$C(n, 0)=1$$

Find the number of two-digit numerals that can be formed using the digits \(2,3,5,6,\) and, 9 and that contain no repeated digits.

Two cards are drawn at random successively from a standard deck. The first card is replaced before the second is drawn. Find the probability that: The first is a club and the second a spade.

Five marbles are selected at random from a bag of seven white and six red marbles. Find the probability of each event. All are white balls.

Solve each equation, where \(n \geq 0\). $$C(n, 1)=10$$

Using the recursive definition of \(b_{n},\) compute each. $$b_{4}$$

A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that: Repeat no letters or digits.

Find the number of three-digit numerals that can be formed using the digits \(2,3,5,6,\) and \(9,\) if repetitions are not allowed.

Five marbles are selected at random from a bag of seven white and six red marbles. Find the probability of each event. All are red balls.

There are five types of desserts available at a restaurant. Find the number of ways eight people can select them, if order does not matter.

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

Solve each equation, where \(n \geq 0\). $$C(n, 2)=28$$

Five marbles are selected at random from a bag of seven white and six red marbles. Find the probability of each event. Three are white and two are red.

The nth Fibonacci number \(F_{n}\) is given by the sum of the numbers along the \(n\) h 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. $$ F_{1} $$

A restaurant offers six choices for the main dish. How many ways can a group of nine women select the main dish? Assume that order does not matter.

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

Solve each equation, where \(n \geq 0\). $$C(n, n-2)=55$$

A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that: Have the property that both words and numbers are palindromes.

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. $$F_{1}$$

The nth Fibonacci number \(F_{n}\) is given by the sum of the numbers along the \(n\) h 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. $$ F_{2} $$

In how many ways can 10 quarters in a piggy bank be distributed among 7 people?

Find the number of ways of dividing a set of size \(n\) into two disjoint subsets of sizes \(r\) and \(n-r\).

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

An old zip code in the United States consists of five digits. Find the total number of possible zip codes that: Have no repetitions.

Five marbles are selected at random from a bag of seven white and six red marbles. Find the probability of each event. Two are white and three are green.

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. $$F_{2}$$

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. $$\\{\mathrm{a}\\}$$

The nth Fibonacci number \(F_{n}\) is given by the sum of the numbers along the \(n\) h 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. $$ F_{5} $$

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

A collection plate contains four nickels, five dimes, and seven quarters. In how many ways can you: Choose three coins?

An old zip code in the United States consists of five digits. Find the total number of possible zip codes that: Begin with 0.

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. $$\\{a\\}$$

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. $$F_{5}$$

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. $$\\{\mathrm{a,b}\\}$$

The nth Fibonacci number \(F_{n}\) is given by the sum of the numbers along the \(n\) h 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. $$ F_{6} $$

Find the number of solutions to each equation, where the variables are nonnegative integers. $$x_{1}+x_{2}+x_{3}+x_{4}=7$$

A collection plate contains four nickels, five dimes, and seven quarters. In how many ways can you: Form a sum of 40 cents?