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Factorial, represented by the symbol \(!\), is a fundamental concept in permutations and combinations. The factorial of a non-negative integer \(n\), denoted by \(n!\), is the product of all positive integers less than or equal to \(n\). It is a way of describing the number of ways to arrange \(n\) different items sequentially. For example:

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

A key fact to remember is that \(0! = 1\). This may seem strange at first, but it fits seamlessly with the other mathematical rules and formulas involving factorials. It is vital to understand factorials well as they form the building blocks for calculating permutations and combinations, which are critical in evaluating expressions such as the one given.

Permutations deal with the arrangement of objects in a specific order. The permutation formula helps determine the total number of possible arrangements. The formula for permutations of \(n\) objects taken \(r\) at a time is given by:\[nPr = \frac{n!}{(n-r)!}\]In this context:

• \(n\) is the total number of objects.
• \(r\) is the number of objects we are choosing.
• The order in which these \(r\) objects are arranged matters.

Let's consider the exercise with \(7P3\): - With \(n = 7\) and \(r = 3\), you calculate it as \(\frac{7!}{(7-3)!} = \frac{7!}{4!}\).- Simplifying gives a product of \(7 \times 6 \times 5 = 210\).This result, 210, signifies the number of different ways you can arrange 3 objects out of a total of 7, where the order is important.

Combinations are used when the order of selection does not matter. The combination formula helps find the number of ways to choose a subset of items from a larger set without regard to the order of the items. The formula for combinations is:\[nCr = \frac{n!}{r!(n-r)!}\]Here's how you can understand and apply it:

• \(n\) is the total number of items available.
• \(r\) indicates how many items you are selecting.
• The key distinction from permutations is that the order of selection does not count here.

In the provided example for \(7C3\):- With \(n = 7\) and \(r = 3\), you calculate \(\frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35\).This result, 35, shows the total number of ways to select 3 items from 7, focusing on selection instead of arrangement. This calculation subtracts the arrangement aspect, leaving only the picking aspect.