Final answer:
To find the value of 5!, multiply 5 by each of the integers that precede it down to 1, which results in 120. Factorials are used in combinatorics and probability theory, such as counting the number of permutations of a set of items.
Step-by-step explanation:
Finding n! for n=5
The factorial of a number n, denoted as n!, is the product of all positive integers from 1 to n. To find the value of 5! without using a calculator, we simply multiply the sequence of whole numbers from 5 down to 1:
- Start with the highest number: 5.
- Multiply it by the next lower number: 5 × 4 = 20.
- Continue the process with the next lower numbers: 20 × 3 = 60, then 60 × 2 = 120, and finally, 120 × 1 = 120.
Therefore, 5! equals 120. This concept is essential in combinatorics and probability theory, where it's often used to count the number of ways to arrange or select items from a set.
Another way to understand factorials is to consider an example of permutations. If you have 5 differently colored balls and you want to know in how many different ways you can arrange them, the answer is 5!, which as we calculated, is 120 different combinations.
Understanding factorials is also crucial for calculating binomial coefficients, which are used to determine the number of ways to choose a subset of items from a larger set, without considering the order of the items.