How many different ways can a director select 4 actors from a group of 20 actors to attend a workshop on performing in rock musicals?

Verified step by step guidance

Recognize that this is a combination problem because the order in which the actors are selected does not matter.

The formula for combinations is given by:

C (n, r) = \frac{n}{!}

, where

n

is the total number of items (actors) and

r

is the number of items to choose (actors to select).

Substitute the values

n = 20

and

r = 4

into the formula:

C (20, 4) = \frac{20}{!}

Simplify the denominator:

20 - 4 = 16

, so the formula becomes:

\frac{20}{!}

Cancel out the common terms in the numerator and denominator by expanding the factorials, leaving only the terms needed to compute the combination. This simplifies the calculation to:

\frac{20 \times 19 \times 18 \times 17}{4}

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Combinations

Combinations refer to the selection of items from a larger set where the order of selection does not matter. In this context, the director is choosing 4 actors from a group of 20, which is a classic example of a combination problem. The formula for combinations is given by C(n, r) = n! / (r!(n - r)!), where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.

Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics, used to calculate the total arrangements or selections of items. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are essential in the combinations formula, as they help determine the number of ways to arrange or select items.

Binomial Coefficient

The binomial coefficient, often represented as C(n, r) or 'n choose r', quantifies the number of ways to choose r elements from a set of n elements without regard to the order of selection. It is calculated using the formula C(n, r) = n! / (r!(n - r)!). This concept is crucial for solving problems involving selections, such as determining how many different groups of actors can be formed.

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