Chapter 6

Find the number of lines that can be drawn using 10 distinct points, no three being collinear.

Find the number of terms in the expansion of each expression. $$(b+c)(d+e+f)(x+y+z)$$

Find the number of bytes that: Contain exactly eight 0's.

Mark each sentence as true or false, where \(n\) is an arbitrary nonnegative integer and \(0 \leq r \leq n\). $$5 \cdot 4 !=5 !$$

Using the alternate inclusion-exclusion formula, find the number of primes not exceeding: 110

The Sealords have three children. Assuming that the outcomes are equally likely and independent, find the probability that they have three boys, knowing that: The first two children are boys.

Using the binomial theorem, expand each. $$(x+2 y)^{6}$$

Find the number of bytes that: Contain exactly nine \(0^{\prime} \mathrm{s}\) .

Find the number of triangles that can be drawn using 10 points, no three being collinear.

Find the number of terms in the expansion of each expression. $$\left(\sum_{i=0}^{2} a_{i}\right)\left(\sum_{i=1}^{4} b_{i}\right)\left(\sum_{i=2}^{5} c_{i}\right)$$

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

Two dice are rolled. Find the probability of obtaining: A sum less than five.

Using the alternate inclusion-exclusion formula, find the number of primes not exceeding: 125

Find the middle term in the binomial expansion of each. $$\left(x+\frac{1}{x}\right)^{4}$$

Find the number of ternary words over the alphabet \\{0,1,2\\} that are of length four and: Contain exactly three 0 's.

Solve the recurrence relation \(d_{n}=-d_{n-1}, n \geq 2,\) where \(d_{1}=-1\).

Find the number of terms in the expansion of each expression. $$\left(\sum_{i=-2}^{5} a_{i}\right)\left(\sum_{i=-1}^{3} b_{i}\right)\left(\sum_{i=0}^{4} c_{i}\right)$$

Mark each sentence as true or false, where \(n\) is an arbitrary nonnegative integer and \(0 \leq r \leq n\). $$(2+3) !=2 !+3 !$$

Using the alternate inclusion-exclusion formula, find the number of primes not exceeding: 129

Find the middle term in the binomial expansion of each. $$\left(x-\frac{1}{x}\right)^{6}$$

Find the number of solutions to each equation with non-negative integer variables. \(x+y+z=11, x \leq 3, y \leq 4, z \leq 5\)

Find the number of ternary words over the alphabet \\{0,1,2\\} that are of length four and: Contain at most two \(0^{\prime} s.\)

Prove each. \(D_{n}\) is even if \(n\) is an odd integer.

Let \(A\) be a 10 -element subset of the set \(\\{1,2, \ldots, 20\\}\) Determine if \(A\) has two five-element subsets that yield the same sum of the elements.

Find the number of palindromic alphanumeric identifiers of length \(n\).

Find the middle term in the binomial expansion of each. $$\left(2 x+\frac{2}{x}\right)^{8}$$

Find the number of solutions to each equation with non-negative integer variables. \(w+x+y+z=13, w \leq 3, x, y \leq 4, z \leq 7\)

A survey of 475 customers at Chestnut Restaurant shows that of the three ice cream flavors - chocolate, strawberry, and vanilla -65 customers 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. Find the probability that a customer selected at random from the survey:

Prove each. \(D_{n}\) is odd if \(n\) is an even integer.

Find the middle term in the binomial expansion of each. $$\left(x^{2}+\frac{1}{x^{2}}\right)^{10}$$

Let \(A\) be a 10 -element subset of the set \(\\{1,2, \ldots, 20\\}\) Determine if \(A\) has two eight-element subsets that yield the same sum of the elements.

In \(1984,\) E. T. H. Wang of Wilfrid Laurier University, Waterloo, Ontario, Canada, established that $$\sum_{r=1}^{n} r^{3}\left(\begin{array}{l}n \\\r\end{array}\right) D_{n-r}=5 n !$$ Verify the formula for \(n=5\) and \(n=6\).

A survey of 475 customers at Chestnut Restaurant shows that of the three ice cream flavors - chocolate, strawberry, and vanilla -65 customers 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. Find the probability that a customer selected at random from the survey: Likes vanilla, given that she does not like chocolate.

Find the largest binomial coefficient in the expansion of each. $$(x+y)^{5}$$

(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. Find the number of gifts sent on the 12 th day.

A word over the alphabet \\{0,1,2\\} is called a ternary word. Find the number of ternary words of length \(n\) that can be formed.

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. Find the number of gifts sent on the 12 th day.

A survey of 475 customers at Chestnut Restaurant shows that of the three ice cream flavors - chocolate, strawberry, and vanilla -65 customers 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. Find the probability that a customer selected at random from the survey: Likes chocolate, given that she does not like strawberry or vanilla.

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

Find the largest binomial coefficient in the expansion of each. $$(x+y)^{6}$$

In Exercises \(13-16\) , use Figure 6.17 to find the number of possible routes from A to the given point, traveling easterly or northerly for the given number of blocks. Point \(C\) and 8 blocks.

(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. Find the total number of gifts sent in 12 days.

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: Can be formed.

Mark each sentence as true or false, where \(n\) is an arbitrary nonnegative integer and \(0 \leq r \leq n\). $$P(n, 1)=P(n, n-1)$$

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. Find the total number of gifts sent in 12 days.

A survey of 475 customers at Chestnut Restaurant shows that of the three ice cream flavors - chocolate, strawberry, and vanilla -65 customers 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. Find the probability that a customer selected at random from the survey: Does not like chocolate, given that she does not like strawberry or vanilla.

Find the largest binomial coefficient in the expansion of each. $$(x+y)^{7}$$

Let \(b_{n}\) denote the number of computations (additions and multiplications) needed to find \(D_{n},\) using the recursive definition \((6.1) .\) Compute the following. $$b_{4}$$

(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 number of gifts sent on the \(n\) th day is \(C(n+1,2),\) 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 the word BAT.