Chapter 6

Find the number of ways a committee of five can be formed from a group of five boys and four girls, if each committee must contain: Exactly two boys.

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: Plants of the same family must be next to each other.

Find the number of solutions to each equation. $$x_{1}+x_{2}+x_{3}+x_{4}=10, x_{1}, x_{2} \geq 2, x_{3} \geq 0, x_{4} \geq 5$$

A die is rolled four times. Find the probability of obtaining: All sixes.

Using the binomial theorem, prove each. \(\sum_{r=0}^{n}\left(\begin{array}{c}n \\\ r\end{array}\right)\left(\begin{array}{c}n \\\ n-r\end{array}\right)\left(\begin{array}{c}2 n \\ n\end{array}\right)\) [Hint: Consider \((1+x)^{2 n}=(1+x)^{n}(1+x)^{n} .\) Equate the coefficients of \(\left.x^{n} \text { from either side. }\right]\)

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

Find the number of ways a committee of five can be formed from a group of five boys and four girls, if each committee must contain: At least two boys.

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: The family of zinnias must be in between the other two families.

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

Using the binomial theorem, prove each. \(\sum_{i=1}^{n}\left(\begin{array}{c}n \\\ i-1\end{array}\right)\left(\begin{array}{c}n \\\ i\end{array}\right)=\left(\begin{array}{c}2 n \\ n+1\end{array}\right)\) [Hint: Consider \((1+x)^{2 n}=(x+1)^{n}(1+x)^{n} .\) Equate the coefficients of \(\left.x^{n+1} \text { from both sides. }\right]\)

A die is rolled four times. Find the probability of obtaining: Exactly one six.

Show that \(c_{n}=\mathrm{O}(n)\)

Find the number of ways a committee of five can be formed from a group of five boys and four girls, if each committee must contain: At least two girls.

Find the number of ways seven boys and three girls can be seated in a row if: A boy sits at each end of the row.

Evaluate each sum. \(1\left(\begin{array}{l}n \\ 1\end{array}\right)+2\left(\begin{array}{l}n \\\ 2\end{array}\right)+3\left(\begin{array}{l}n \\\ 3\end{array}\right)+\cdots+n\left(\begin{array}{l}n \\ n\end{array}\right)\) (Hint: Let \(S\) denote the sum. Use \(S\) and the sum in the reverse order to compute \(2 S .\))

A die is rolled four times. Find the probability of obtaining: Exactly two sixes.

Find the number of ways a committee of five can be formed from a group of five boys and four girls, if each committee must contain: At least one boy and at least one girl.

Evaluate each sum. $$ a\left(\begin{array}{l}{n} \\\ {0}\end{array}\right)+(a+d)\left(\begin{array}{l}{n} \\\ {1}\end{array}\right)+(a+2 d)\left(\begin{array}{l}{n} \\\ {2}\end{array}\right)+\cdots+(a+n d)\left(\begin{array}{l}{n} \\\ {n}\end{array}\right) $$ (Hint: Use the same hint as in Exercise \(34 .\) )

A die is rolled four times. Find the probability of obtaining: Exactly three sixes.

Find the number of ways 10 quarters can be distributed among three people \(-\) Aaron, Beena, and Cathy - so that both Aaron and Beena get at least one quarter, Beena gets no more than three, and Cathy gets at least two.

Let \(a_{n}\) denote the number of multiplications needed to compute \(D_{n}\) using the formula in Exercise 34. Show that \(a_{n}=\mathrm{O}\left(n^{2}\right)\).

Find the number of ways a committee of five can be formed from a group of five boys and four girls, if each committee must contain: At most one boy.

Find the number of ways seven boys and three girls can be seated in a row if: The girls sit together at one end of the row.

Evaluate each sum. Show that \(C(n, r-1) < C(n, r)\) if and only if \(r < \frac{n+1}{2},\) where \(0 \leq r < n .\)

Find the number of ways 11 raisins can be distributed among four children - Daisy, Emily, Francis, Gail-so that Daisy, Emily, and Francis get at least two raisins, Francis gets no more than four, and Gail gets at least three.

Find the number of ways a committee of five can be formed from a group of five boys and four girls, if each committee must contain: At most one girl.

Find the number of ways seven boys and three girls can be seated in a row if: Show that \(P(n, 0)=1\)

A die is rolled four times. Find the probability of obtaining: At least one six.

Evaluate each sum. Show that \(C(n, r-1) < C (n, r)\) if and only if \(r < \frac{n+1}{2},\) where \(0 \leq r < n .\)

A die is rolled four times. Find the probability of obtaining: Not more than two sixes.

Using the recursive definition of \(P(n, r),\) evaluate each. $$P(5,4)$$

Verify that \(\left(\begin{array}{l}n \\\ r\end{array}\right)=\frac{n}{r}\left(\begin{array}{l}n-1 \\\ r-1\end{array}\right),\) where \(n \geq r \geq 1\)

Using induction, prove each. $$\left(\begin{array}{l}n \\ 0\end{array}\right)+\left(\begin{array}{c}n+1 \\\ 1\end{array}\right)+\left(\begin{array}{c}n+2 \\\ 2\end{array}\right)+\cdots+\left(\begin{array}{c}n+r \\\ r\end{array}\right)=\left(\begin{array}{c}n+r+1 \\ r\end{array}\right) \text { (Hint: Use Pascal's identity.) }$$

A survey shows that \(20 \%\) of the adults in Simpleton have high blood pressure. A sample of four adults is selected at random. Find the probability that: They all have high blood pressure.

Find the number of bytes that: Have the same third and fourth bits.

Using the recursive definition of \(P(n, r),\) evaluate each. $$P(6,0)$$

Using induction, prove each. $$1\left(\begin{array}{l}n \\ 1\end{array}\right)+2\left(\begin{array}{l}n \\\ 2\end{array}\right)+\cdots+n\left(\begin{array}{l}n \\\ n\end{array}\right)=n 2^{n-1}$$

A survey shows that \(20 \%\) of the adults in Simpleton have high blood pressure. A sample of four adults is selected at random. Find the probability that: Exactly one of them has high blood pressure.

Prove Pascal's identity algebraically.

Using the recursive definition of \(P(n, r),\) evaluate each. $$P(3,2)$$

Using induction, prove each. $$\left(\begin{array}{l}n \\ 0\end{array}\right)^{2}+\left(\begin{array}{l}n \\\ 1\end{array}\right)^{2}+\left(\begin{array}{l}n \\\ 2\end{array}\right)^{2}+\cdots+\left(\begin{array}{l}n \\\ n\end{array}\right)^{2}=\left(\begin{array}{l}2 n \\ n\end{array}\right)$$

A survey shows that \(20 \%\) of the adults in Simpleton have high blood pressure. A sample of four adults is selected at random. Find the probability that: Not more than two of them have high blood pressure.

Using the recursive definition of \(P(n, r),\) evaluate each. $$P(6,3)$$

From the binomial expansion \((1+x)^{n}=\sum_{r=0}^{n}\left(\begin{array}{l}{n} \\\ {r}\end{array}\right) x^{r},\) it can be shown using calculus that \(n(1+x)^{n-1}=\sum_{r=1}^{n}\left(\begin{array}{c}{n} \\\ {r}\end{array}\right) r x^{n-1}\) . Using this result, prove each. $$ 1\left(\begin{array}{l}{n} \\\ {1}\end{array}\right)+2\left(\begin{array}{l}{n} \\\ {2}\end{array}\right)+3\left(\begin{array}{l}{n} \\\ {3}\end{array}\right)+\cdots+n\left(\begin{array}{l}{n} \\\ {n}\end{array}\right)=n 2^{n-1} $$

A survey shows that \(20 \%\) of the adults in Simpleton have high blood pressure. A sample of four adults is selected at random. Find the probability that: Not all of them have high blood pressure.

Solve each equation. $$P(n, 1)=6$$

From the binomial expansion \((1+x)^{n}=\sum_{r=0}^{n}\left(\begin{array}{l}n \\\ r\end{array}\right) x^{r},\) it can be shown using calculus that \(n(1+x)^{n-1}=\sum_{r=1}^{n}\left(\begin{array}{c}n \\ r\end{array}\right) r x^{r-1} .\) Using this result, prove each. $$1\left(\begin{array}{l}n \\ 1\end{array}\right)+3\left(\begin{array}{l}n \\\ 3\end{array}\right)+5\left(\begin{array}{l}n \\\ 5\end{array}\right)+\cdots=2\left(\begin{array}{l}n \\\ 2\end{array}\right)+4\left(\begin{array}{l}n \\\ 4\end{array}\right)+6\left(\begin{array}{l}n \\ 6\end{array}\right)+\cdots=n 2^{n-2}$$

Using the explicit formula in Example \(6.26,\) verify that \(g_{n}=\) \(\sum_{k=0}^{4} C(n-1, k)\).

Solve each equation. $$P(n, 2)=42$$

For the casino game football pools, a list of 10 football games is printed on a ticket. If one team is considered weaker than its opponent by the people who run the pool, that team is given enough points to make the game a tossup. Thus the probability of picking a winning team is 0.5. You pay \(\$ 1\) to play the game and select all 10 winners. If all your selections win, you get \(\$ 150 ;\) if nine win, you receive a consolation prize of \(\$ 20 ;\) otherwise, you lose your dollar. Compute your expected profit. (A. Sterrett, 1967 )