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?

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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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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.

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Combinations

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.

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Factorials

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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