Permutations, Combinations, and the Binomial Theorem

 Permutations are used to count the number of ways items can be arranged when order matters.

Permutations: order matters.

 Combinations are used to count the number of ways items can be chosen when order does not matter.

Combinations: order does not matter. 

 The Binomial Theorem is a shortcut for expanding binomials raised to a power. It uses patterns and combinations to determine each term without multiplying step by step. 

Binomial Theorem: a shortcut for expanding binomials.

 A binomial is an expression with two terms. The Binomial Theorem helps expand binomials raised to a power without multiplying them repeatedly.
Examples of binomials:
 "x + 3"

"2x − 5"

"a + b"

What do these have in common?
They all contain two terms.
(A term can be a number, a variable, or both.)

The Binomial Theorem is used when a binomial is raised to a power, such as:

(x+3)⁴

Problem:
Three students — Alex, Jordan, and Sam — are lining up for a photo.
How many different line-ups are possible?

Why this is a permutation:

• The order matters
   
• Alex–Jordan–Sam is different from Sam–Jordan–Alex
   

Because changing the order creates a new result, this is a permutation.

(Note: The specific problem details are not as important as deciding whether the situation is a permutation or a combination.)

 Problem:

A teacher needs to choose 3 students from a class of 10 to be on a committee.

How many different committees are possible?

Why this is a combination:

• Only which students are chosen matters
   
• The order they are listed in does not matter
   
• A committee with Alex, Jordan, and Sam is the same no matter the order
   

Since the order does not matter, this is a combination problem.

(Note: The specific problem details are not as important as deciding whether the situation is a permutation or a combination.)