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10. Combinatorics & Probability

Combinatorics

10. Combinatorics & Probability

Combinatorics

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Fundamental Counting Principle

Fundamental Counting Principle

Introduction to Permutations

Introduction to Permutations

Permutations of Non-Distinct Objects

Permutations of Non-Distinct Objects

Permutations vs. Combinations

Permutations vs. Combinations

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Practice this topic

Multiple Choice

How many possible outcomes are there if you roll 5 dice?

Multiple Choice

How many options are there for license plates with any three letters (A-Z) followed by any 3 numbers (0-9)?

Multiple Choice

Phone numbers are 10 digits long. How many possible phone numbers are there if the 1^st and 4^th numbers can't be 0?

Multiple Choice

A student formed a club at their school. They have 13 members, and need to elect a president, vice president, and treasurer. How many ways are there to fill these officer positions?

Multiple Choice

Emily is organizing her closet. She has 15 shirts left to hang but has space in one section for 6 shirts. How many ways could she hang shirts in that section?

Multiple Choice

Evaluate the given expression. $sub 9 P sub 4$

Multiple Choice

You want to arrange the books on your bookshelf by color. How many different ways could you arrange 12 books if 4 of them have a blue cover, 3 are yellow, and 5 are white?

Multiple Choice

How many ways are there to arrange the letters in the word CALCULUS?

Multiple Choice

From a class of 28 students, in how many ways could a teacher select 4 students to lead the class discussion?

Multiple Choice

Evaluate the given expression. $sub 11 C sub 7$

Textbook Question

In Exercises 39–48, find the term indicated in each expansion. (x − 1/2)^9; fourth term

Textbook Question

In Exercises 39–48, find the term indicated in each expansion. (x²+y³)^8; sixth term

Textbook Question

In Exercises 39–48, find the term indicated in each expansion. (x − 1)^9; fifth term

Textbook Question

In Exercises 39–48, find the term indicated in each expansion. (2x + y)^6; third term

Textbook Question

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (y³ − 1)^20

Textbook Question

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (x² + 1)^16

Textbook Question

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (x - 2y)^10

Textbook Question

In Exercises 31–38, write the first three terms in each binomial expansion, expressing the result in simplified form. (x+2)^8

Textbook Question

Use the Binomial Theorem to expand and then simplify the result: (x² +x+ 1)³.

Textbook Question

Find the middle term in the expansion of (3/x + x/3)^10

Textbook Question

n Exercises 53–54, find f(x + h) − f(x)/h and simplify. f(x) = x^4+7

Textbook Question

In Exercises 49–52, use the Binomial Theorem to expand each expression and write the result in simplified form. (x^1/3 +x^-1/3)^3

Textbook Question

In Exercises 49–52, use the Binomial Theorem to expand each expression and write the result in simplified form. (x^3 +x^-2)^4

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (2a + b)^6

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (3x − y)^5

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (x − 1)^5

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (c+2)^5

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (2x³ − 1)^4

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (y-3)^4

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (x²+2y)^4

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (2x + 1)^4

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (5x – 1)^3

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (3x + y)^3

Textbook Question

In Exercises 9–30, use the Binomial Theorem to expand each binomial and express the result in simplified form. (x+2)³

Textbook Question

In Exercises 1–8, evaluate the given binomial coefficient.

Textbook Question

In Exercises 1–8, evaluate the given binomial coefficient.

Textbook Question

In Exercises 1–8, evaluate the given binomial coefficient.

Textbook Question

In Exercises 1–8, evaluate the given binomial coefficient.

Textbook Question

Write the first three terms in the binomial expansion, expressing the result in simplified form. (x-3)^9

Textbook Question

Use the Binomial Theorem to expand the binomial and express the result in simplified form. ((x^2)-1)^4

Textbook Question

Use the Binomial Theorem to expand the binomial and express the result in simplified form. (2x+1)^3

Textbook Question

Find the term indicated in the expansion. (2x-3)^6; fifth term

Textbook Question

Evaluate the given binomial coefficient 11 8

Textbook Question

Find the term in the expansion of (x² + y²)^5 containing x^4 as a factor.

Textbook Question

Exercises 89–91 will help you prepare for the material covered in the next section. Evaluate n!/(n-r)! for n = 20 and r = 3

Textbook Question

Exercises 89–91 will help you prepare for the material covered in the next section. Evaluate n!/(n-r)!r! for n = 8 and r = 3

Textbook Question

How many four-digit odd numbers less than 6000 can be formed using the digits 2, 4, 6, 7, 8, and 9?

Textbook Question

Use the formula for nCr to solve Exercises 49–56. To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?

Textbook Question

Use the formula for nCr to solve Exercises 49–56. You volunteer to help drive children at a charity event to the zoo, but you can fit only 8 of the 17 children present in your van. How many different groups of 8 children can you drive?

Textbook Question

Use the formula for nCr to solve Exercises 49–56. Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?

Textbook Question

Use the formula for nCr to solve Exercises 49–56. An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?

Textbook Question

Use the Fundamental Counting Principle to solve Exercises 29–40. An ice cream store sells two drinks (sodas or milk shakes) in four sizes (small, medium, large, or jumbo) and five flavors (vanilla, strawberry, chocolate, coffee, or pistachio). In how many ways can a customer order a drink?

Textbook Question

Use the Fundamental Counting Principle to solve Exercises 29–40. A popular brand of pen is available in three colors (red, green, or blue) and four writing tips (bold, medium, fine, or micro). How many different choices of pens do you have with this brand?

Textbook Question

Use the Fundamental Counting Principle to solve Exercises 29–40. The model of the car you are thinking of buying is available in nine different colors and three different styles (hatchback, sedan, or sport). In how many ways can you order the car?

Textbook Question

In Exercises 21–28, evaluate each expression.

Textbook Question

In Exercises 21–28, evaluate each expression.

Textbook Question

In Exercises 21–28, evaluate each expression.

Textbook Question

In Exercises 21–28, evaluate each expression.

Textbook Question

In Exercises 9–16, use the formula for nCr to evaluate each expression. 5C0

Textbook Question

In Exercises 9–16, use the formula for nCr to evaluate each expression. 7C7

Textbook Question

In Exercises 9–16, use the formula for nCr to evaluate each expression. 11C4

Textbook Question

In Exercises 1–8, use the formula for nPr to evaluate each expression. 8P0

Textbook Question

In Exercises 1–8, use the formula for nPr to evaluate each expression. 6P6

Textbook Question

In Exercises 1–8, use the formula for nPr to evaluate each expression. 8P5

Textbook Question

In Exercises 1–8, use the formula for nPr to evaluate each expression. 9P4

Textbook Question

In how many ways can five airplanes line up for departure on a runway?

Textbook Question

How many different ways can a director select 4 actors from a group of 20 actors to attend a workshop on performing in rock musicals?

Textbook Question

A club with 15 members is to choose four officers–president, vice president, secretary, and treasurer. In how many ways can these offices be filled?

Textbook Question

Evaluate the expression. *permutation notation* the number of permutations 9 things taken 5 at a time (sub 9)P(sub 5)

Textbook Question

Evaluate the expression. *permutation notation* the number of permutations 8 things taken 3 at a time (sub 8)P(sub 3)

Textbook Question

In Exercises 9–16, use the formula for nCr to evaluate each expression. 9C5

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