Flashcard on Combinatorics and Number Theory

Front

What is Combinatorics?

Back

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

Front

Define a natural number.

Back

A natural number is a positive integer, typically starting from 1, used for counting.

Front

What is the principle of counting?

Back

The principle of counting 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.

Front

How do you determine if a number is divisible by 5?

Back

A number is divisible by 5 if its last digit is either 0 or 5.

Front

What is a prime factor?

Back

A prime factor is a factor that is a prime number, meaning it has no positive divisors other than 1 and itself.

Front

How do you find the number of positive divisors of a number?

Back

To find the number of positive divisors, factor the number into its prime factors, then use the formula: (e1 + 1)(e2 + 1)...(en + 1), where e1, e2,..., en are the exponents in the prime factorization.

Front

What is a combination in combinatorics?

Back

A combination is a selection of items from a larger set where the order does not matter.

Front

What is a permutation in combinatorics?

Back

A permutation is an arrangement of items from a set where the order does matter.

Front

How many ways can you choose 2 items from a set of 5?

Back

The number of ways to choose 2 items from a set of 5 is calculated using the combination formula: C(5,2) = 10.

Front

What is the formula for combinations?

Back

The formula for combinations is C(n, r) = n! / (r!(n-r)!), where n is the total number of items, and r is the number of items to choose.