Waste less time on Facebook — follow Brilliant.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Combinations
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Concept Quizzes
Suppose \(5\) distinct numbers are chosen from nine integers \(1, 2, \ldots, 9\) to create a \(5\)-digit number. Given that the digits chosen must consist of \(3\) odd numbers and \(2\) even numbers, how many distinct \(5\)-digit numbers can be created?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
3 red balls and \(12\) blue balls are randomly placed in a row. What is the probability that no two red balls are adjacent?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Suppose a \(4\)-digit number is created from \(4\) distinct integers chosen among \(\{1, 2, 3, 4, 5, 7, 9\}.\) How many such \(4\)-digit numbers include both \(1\) and \(2\) but neither begin nor end with \(1?\)
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
In Sue's Fruit & Flowers Market, there are \(8\) different kinds of fruit, including peaches and apples. There are also \(6\) different kinds of flowers, including roses and lilies. How many ways are there to make a gift basket such that the basket has \(3\) different kinds of fruit including peaches and \(2\) different kinds of flowers with no roses?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
From a group of \(11\) students including \(A\) and \(B\), \(4\) people are selected to be arranged in a line. If both \(A\) and \(B\) must be among the \(4\) selected students, how many distinct line-ups are possible?