Question

Find the value of the following factorial expressions: \((a) 5 !\) (b) \(8 !\) (c) \(\frac{4 !}{3 !}\) (d) \(\frac{6 !}{4 !}\) \((e) \frac{(n+2) !}{n !}\)

Short answer

(a) 120, (b) 40,320, (c) 4, (d) 30, (e) \((n+2) \times (n+1)\).

Step-by-step solution

Calculate Basic Factorials

To find the value of a factorial like - For (a), calculate \(5!\):\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\]- For (b), calculate \(8!\):\[8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40,320\]These complete parts (a) and (b) of the problem.

Simplify Fractional Factorials

For expressions like (c) - Calculate \(\frac{4!}{3!}\):\[\frac{4!}{3!} = \frac{4 \times 3 \times 2 \times 1}{3 \times 2 \times 1} = 4\]This operation simplifies by canceling out the same terms \((3! = 3 \times 2 \times 1)\) in the numerator and denominator, leaving us with \(4\).Similarly for (d)- Calculate \(\frac{6!}{4!}\):\[\frac{6!}{4!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} = 6 \times 5 = 30\]This time, the terms from \(4!\) cancel out, so only \(6 \times 5\) remain.

Generalize for Variable Factorials

For expressions like (e)- Calculate \(\frac{(n+2)!}{n!}\):\[(n+2)! = (n+2) \times (n+1) \times n! \\frac{(n+2)!}{n!} = \frac{(n+2) \times (n+1) \times n!}{n!} = (n+2) \times (n+1)\]In this case, the \(n!\) in the numerator and denominator cancel each other, leaving the expression simplified to \((n+2) \times (n+1)\).

Key concepts

Factorial expressions

Factorials are a way to represent products of sequential whole numbers. You often see them written with an exclamation mark, like \(!5\!\). This reads as "five factorial" and it means you multiply all whole numbers from 5 down to 1. So, \(!5\!\) is actually \(5 \times 4 \times 3 \times 2 \times 1\). Each of these numbers is a term in the expression. Factorials grow rapidly larger with bigger numbers because many numbers are multiplied together. This is useful for solving problems in statistics, algebra, and other mathematical fields.
Whenever you encounter a factorial, remember the sequence of multiplications. This pattern is key in transforming difficult math problems into manageable solutions.

Simplifying fractions

Simplifying fractions involving factorials might seem tricky, but with a little practice, you'll get the hang of it. The idea is to cancel common factors from the numerator and the denominator.

For example, consider the fraction \(\frac{4!}{3!}\). You start by writing out both factorials:

• \(4! = 4 \times 3 \times 2 \times 1\)
• \(3! = 3 \times 2 \times 1\)

You'll notice that \(3!\) is part of \(4!\). This allows you to cancel out these terms in both the numerator and the denominator. What remains in the numerator is just \(4\), making the simplified expression \(4\).
Remember to always look for such common terms when simplifying fractions with factorials. This can make even the largest factorial fractions much more digestible.

Mathematical operations

Mathematics is full of operations, and factorials make good use of multiplication as their core operation. But handling them often involves additional operations like division, especially when dealing with factorial fractions.

When simplifying factorial fractions, you're essentially dividing one product of numbers by another. The trick is to figure out which numbers can be canceled due to being present in both the numerator and the denominator. So, the division is more about simplifying the terms rather than a straightforward "/" operation.
Practice dividing factorials to hone this skill, and soon you'll be cancelling terms with ease and reducing large expressions to simple and manageable results.

Basic factorial calculations

When beginning with factorial calculations, it helps to first master the basics. Calculating small factorials manually assists in understanding how they expand. Let's recap: \(1!\) is \(1\), \(2!\) is \(2 \times 1 = 2\), and \(3!\) is \(3 \times 2 \times 1 = 6\), and so on.

For increasing numbers, the calculations become larger:

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)
• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)
• And \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\)

It’s insightful to calculate a few factorials by hand to fully grasp the process. This way, when you encounter a factorial in tougher mathematical problems, the understanding and recognition are immediate.