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What is 16 factorial ?
Steps to calculate factorial of 16
To find 16 factorial, or 16!, simply use the formula that multiplies the number 16 by all positive whole numbers less than it.
Let’s look at how to calculate the Factorial of 16:
Factorials of Numbers similar to 16
What is Factorial?
A factorial is a mathematical function represented by an exclamation point (!) following a number, indicating the product of all positive integers from 1 to that number. When considering the factorial of 16, denoted as 16!, it signifies a significant value in combinatorial mathematics, allowing us to calculate permutations and combinations in scenarios with 16 different elements, making it a vital concept in understanding complex mathematical problems.
Formula to Calculate the Factorial of 16
The procedure to find the factorial of any number, including 16, follows the basic formula n! = n × (n-1) × … × 2 × 1. Applying this to 16, we get:
16! = 16 × 15 × 14 × … × 2 × 1. Calculating it step by step, we begin with 16 and sequentially multiply it by each preceding integer:
16 × 15 × 14 × … × 3 × 2 × 1 = 20,922,789,888,000 — an impressively large number representing the vast number of permutations of 16 unique items.
What is the Factorial of 16 Used For?
Factorials, particularly large ones like 16!, are used extensively in fields requiring computation of permutations and combinations. They play a critical role in probability theory, to determine the probability of an event occurring within a defined number of possibilities. In combinatorics, 16! can be used to calculate how many ways we can arrange or select items from a set of 16, which is critical in the organization of tournaments, designing computer algorithms or encryption systems, and solving complex scheduling problems.
Exercises
- Calculate the number of zeros at the end of the value of 16!.
- If a team has 16 members, in how many ways can a captain and vice-captain be chosen?
- Without calculating the exact number, determine if 16! is an even or odd number.
Solutions to Exercises
- The value of 16! ends with 3 zeros. (Because there are three pairs of 5 and 2 within the factors of 16! which multiplies out to be 10).
- For a team of 16 members, the number of ways to choose a captain and vice-captain is 16! / (16-2)! = 16 × 15 = 240 ways.
- 16! is an even number, as its calculation includes the multiplication by the even number 2 (among others).
Frequently Asked Questions
Q1: What can you say about the magnitude of 16! ?
The factorial of 16 is a very large number, specifically 20,922,789,888,000. Its magnitude reflects the massive number of different ways to arrange or select items from a set of 16 entities.
Q2: Can you explain why factorials of even numbers are always even?
Factorials of even numbers are always even because the product includes at least one instance of the number 2, which is the smallest even prime number. Since multiplication by an even number results in an even product, the factorial of an even number must also be even.
Other conversions of the number 16
16 in Roman numerals
16 in Spanich
16 in Italian