91Ó°ÊÓ

To get a solid understanding of permutations, you need to grasp the concept of factorials first. A factorial, symbolized as \(!\), is the product of all positive integers up to a specific number. For example, the factorial of 5, written as \(5!\), is calculated as:

5! = 5 \times 4 \times 3 \times 2 \times 1 = 120

Factorials grow incredibly fast. Take 6!, which we calculated in the original exercise:
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720

Knowing how to compute factorials is crucial when dealing with both permutations and combinations. They form the backbone of these concepts.

The permutation formula is essential when you're dealing with arrangements where order matters. The formula to find permutations is:

\[ _n P_{r} = \frac{n!}{(n-r)!} \ \]
For the problem \(6 P_2\), we identified that:

* n is 6 (total items)
* r is 2 (number of items to choose)

Using the permutation formula, we substitute \(n = 6\) and \(r = 2\):

\[ _6 P_{2} = \frac{6!}{(6-2)!} = \frac{720}{24} = 30 \]
It's crucial to recognize that permutations are all about order. For instance, selecting and arranging 2 items out of 6 items yields 30 unique sequences.

• Step 1: Calculate \(n!\) (6!)
• Step 2: Calculate \((n-r)!\) (4!)
• Step 3: Substitute into the formula and solve

While permutations consider the order of items, combinations do not. Combinations simply count how many ways you can choose items from a group, order doesn't matter. The formula for combinations is:

\[ _n C_{r} = \frac{n!}{r!(n-r)!} \ \]
Notice the difference? In combinations, we also divide by \(r!\) because the order is not important. Let's say you wanted to choose 2 items out of 6 and you didn't care about the order. You'd use the combination formula:

\[ _6 C_{2} = \frac{6!}{2!(6-2)!} = \frac{720}{2 \times 24} = 15 \]
So, there are 15 ways to pick 2 items out of 6 in any order. This is helpful in scenarios where you only care about the selection itself, not the arrangement.