What is 6 factorial ?

A factorial, written as ‘n!’, is the product of all positive integers up to a given number ‘n’. It’s a mathematical operation with extensive applications in fields such as algebra, combinatorics, and statistics. The factorial of 6, often signified as 6!, is particularly noteworthy because it marks the transition from relatively simple, smaller factorials to more complex, larger ones, serving as a foundational concept in many mathematical equations and problems.

The general formula for calculating a factorial, ‘n!’, is n × (n-1) × … × 1. To find the factorial of 6:

• Start with the number 6.
• Multiply 6 by one less than itself, which is 5, to get 30.
• Continue this process down to 1: multiply 30 by 4 to get 120, multiply 120 by 3 to get 360, multiply 360 by 2 to get 720, and finally, multiply 720 by 1 to end up with 720.

Thus, the factorial of 6 (6!) is equal to 720.

The factorial of a number has several fascinating uses. The factorial of 6, or 6!, is particularly used in:

• Combinatorics, to calculate permutations and combinations, helping to determine the number of possible ways to arrange or select 6 distinct items.
• Probability theory, where it’s pivotal in calculating probabilities of events in a finite sample space.
• Algebra, for expanding expressions involving exponents and series.

These applications make 6! an essential concept in both theoretical and applied mathematics.

Test your understanding of the factorial of 6 with these exercises:

1. Calculate the number of ways you can arrange 6 different books on a shelf.
2. If you have 6 different pairs of socks, in how many ways can you pick 2 pairs to take on a trip?
3. How many different two-digit numbers can you form using the digits 1 through 6 if each digit must be unique?

Here are the solutions to the exercises provided:

• The number of ways to arrange 6 books is simply 6! (factorial of 6), which equals 720 ways.
• Choosing 2 pairs from 6 gives us a combination problem, calculated by the formula 6! / (2!(6 – 2)!), which equals 15 ways.
• The number of unique two-digit numbers is a permutation of 6 choosing 2, calculated by 6! / (6 – 2)!, which equals 30 numbers.