Evaluate 5 factorial.
5! = 5 × 4 × 3 × 2 × 1 = 120
The quantity 0! has no intuitive meaning, but for consistency it will be defined as 1.
Another way of writing nkP is in terms of factorials. Specifically, from Definition 4.8 we can re-express n kP in the form
The Correct Answer and Explanation is :
The formula for permutations, denoted as ( P(n, k) ), is given by:
[
P(n, k) = \frac{n!}{(n – k)!}
]
where:
- ( n! ) (n factorial) is the product of all positive integers up to ( n ),
- ( (n – k)! ) is the factorial of ( (n – k) ), ensuring we select only ( k ) elements.
Example Calculation:
For instance, if we want to find ( P(5, 3) ), we calculate:
[
P(5,3) = \frac{5!}{(5-3)!} = \frac{5!}{2!}
]
[
= \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = \frac{120}{2} = 60
]
So, the number of ways to arrange 3 objects from a set of 5 is 60.
Explanation:
- Concept of Factorials: The factorial of a number ( n ) represents the total ways to arrange ( n ) distinct objects. The recursive definition is:
[
n! = n \times (n-1)!
]
with 0! defined as 1 by convention. - Permutation Formula: A permutation involves arranging objects in a specific order. The formula arises from considering all ( n! ) arrangements and then dividing by redundant ( (n-k)! ) arrangements of unchosen elements.
- Applications: This formula is useful in probability, statistics, and combinatorics, particularly in scenarios where order matters, such as seating arrangements, passwords, and race rankings.
Image Representation:
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