Factorial

Definition

A factorial, denoted by the symbol '!', is the product of all positive integers up to a specified number. For any positive integer n, the factorial is expressed as n! = n × (n - 1) × (n - 2) × ... × 3 × 2 × 1, with the special case that 0! = 1. Factorials are essential in permutations and combinations, helping to count arrangements and selections.

5 Must Know Facts For Your Next Test

1. The factorial of a number grows rapidly; for example, 5! = 120, while 10! = 3,628,800.
2. Factorials are used in calculating probabilities and distributions in statistics, particularly in problems involving permutations and combinations.
3. The value of n! is defined for non-negative integers only; negative integers do not have a factorial value.
4. Factorials can be computed recursively: n! = n × (n - 1)! with the base case of 0! = 1.
5. The factorial function is also used in calculus, particularly in Taylor series expansions and in defining certain probability distributions.

Review Questions

• How can you apply the concept of factorial to calculate permutations, and why is it important?
  • To calculate permutations, which are arrangements where order matters, you use the factorial of the total number of items being arranged. The formula for permutations is P(n, r) = n! / (n - r)!, where n is the total number of items and r is the number of items to arrange. This shows how many ways you can choose and order r items from n, emphasizing the significance of factorials in counting distinct arrangements.
• In what scenarios would you use combinations instead of permutations when applying factorials?
  • You would use combinations when the order of selection does not matter. For example, if you're choosing a committee from a group, it doesn't matter who is chosen first or last; you only care about who is on the committee. The formula for combinations is C(n, r) = n! / [r! × (n - r)!], which reflects that you divide by the factorial of the selected items to eliminate different orderings that result in the same group.
• Evaluate how understanding factorials enhances your ability to solve complex probability problems involving multiple outcomes.
  • Understanding factorials allows you to break down complex probability problems into manageable parts by calculating arrangements and selections efficiently. When faced with problems that involve multiple outcomes or choices, such as determining possible event sequences or group formations, factorials provide a systematic way to count these possibilities accurately. This foundational knowledge supports deeper statistical analyses and helps to ensure precise results in probability assessments.

Related terms

Permutation:

An arrangement of objects in a specific order, where the order of selection is important.

Combination:

A selection of objects where the order does not matter, typically calculated using factorials.

Binomial Coefficient: A number that represents the number of ways to choose a subset of items from a larger set, often expressed as 'n choose k' or C(n, k), and calculated using factorials.