Counting principle, factorials, permutations, and combinatio

Front

What is the counting principle in combinatorics?

Back

The counting principle states that if there are 'm' ways to do one thing and 'n' ways to do another, then there are m × n ways to do both.

Front

Define factorial and provide an example.

Back

A factorial, denoted as n!, is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

Front

What is a permutation?

Back

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 n! / (n - r)!.

Front

What is a combination?

Back

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 n! / (r! (n - r)!).

Front

How many different sandwiches can be made with 4 ingredients if 2 are chosen at a time?

Back

Using combinations, the number of different sandwiches is 4C2 = 6.

Front

If a family of 3 is to be seated in 3 seats, how many arrangements are possible?

Back

The number of arrangements is 3! = 6.

Front

How do you calculate the number of combinations of pizza with one topping if there are 4 sizes, 2 crust types, and 8 toppings?

Back

The total combinations are calculated as 4 (sizes) × 2 (crusts) × 8 (toppings) = 64.

Front

What is the difference between permutations and combinations?

Back

Permutations consider the order of selection, while combinations do not.

Front

In the context of job applicants, when is a combination used instead of a permutation?

Back

A combination is used when the order of selection does not matter, such as selecting applicants for positions.

Front

How many ways can you select a triple-scoop ice-cream cone with 18 flavors if the order does not matter?

Back

Using combinations with repetition, the number of ways is calculated as (n + r - 1)C(r), where n is the number of flavors and r is the scoops. For 18 flavors and 3 scoops, it is (18 + 3 - 1)C(3) = 816.