A permutation is an arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is given by: $$P(n, r) = \frac{n!}{(n-r)!}$$ .

A combination is a selection of objects without regard to the order. The number of combinations of n objects taken r at a time is given by: $$C(n, r) = \frac{n!}{r!(n-r)!}$$ .

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers up to n. For example, $$5! = 5 × 4 × 3 × 2 × 1 = 120$$ .

How many different models of a car can be purchased if there are 9 colors, 3 styles, and 2 sizes?

The total number of different models is calculated by multiplying the options: $$9 × 3 × 2 = 54$$ .

How many ways can you choose 3 hearts and 1 spade from a 52-card deck?

The number of ways is given by: $$C(13, 3) × C(13, 1) = 286 × 13 = 3718$$ .

What is the result of $$_nC_n$$ ?

The result is 1, as there is only one way to choose all n items from n.

How many ways can 3 officers be chosen from 10 members in a ski club?

The number of ways to choose 3 officers (captain, co-captain, secretary) is given by: $$P(10, 3) = 10 × 9 × 8 = 720$$ .

What is the Counting Principle?

The Counting Principle states that if one event can occur in m ways and a second can occur independently in n ways, then the two events can occur in m × n ways.

What is the Binomial Theorem?

The Binomial Theorem describes the algebraic expansion of powers of a binomial. It states that: $$(a + b)^n = \frac{n!}{k!(n-k)!} a^{n-k} b^k$$ for k = 0, 1, ..., n.

What is the significance of the Pascal's Triangle in Combinatorics?

Pascal's Triangle is a triangular array that represents the coefficients of the binomial expansion and provides a quick way to calculate combinations.

Combinatorics Flashcards

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14 questions

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1.

FLASHCARD QUESTION

Front

What is Combinatorics?

Back

Combinatorics is a branch of mathematics dealing with combinations, arrangements, and counting of objects.

2.

FLASHCARD QUESTION

Front

Define Permutation.

Back

3.

FLASHCARD QUESTION

Front

Define Combination.

Back

4.

FLASHCARD QUESTION

Front

What is the difference between Permutation and Combination?

Back

Permutation considers the order of selection, while combination does not.

5.

FLASHCARD QUESTION

Front

What is a Factorial?

Back

6.

FLASHCARD QUESTION

Front

What is a Tree Diagram?

Back

A tree diagram is a graphical representation used to illustrate all possible outcomes of a situation, often used in probability and combinatorics.

7.

FLASHCARD QUESTION

Front

What is a Sample Space?

Back

A sample space is the set of all possible outcomes of a probabilistic experiment.

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