Permutations

• Permutations are defined as the distinct arrangements of a collection of items where the sequence or position of each item matters. If changing the order of two items creates a new outcome, the scenario is a permutation.

• The Factorial operation, denoted by $n!$, is the foundation of permutation calculations. It represents the product of all positive integers from $n$ down to $1$, such as $4! = 4 \times 3 \times 2 \times 1 = 24$.

• For a set of $n$ unique objects, there are exactly $n!$ ways to arrange them in a row. This is because there are $n$ choices for the first position, $n-1$ for the second, and so on until only one object remains.

• A special case in factorial math is $0! = 1$. This is defined this way because there is exactly one way to arrange zero items: by doing nothing.

• The Fundamental Counting Principle states that if one event can occur in $p$ ways and a second event in $q$ ways, the total number of ways both can occur is $p \times q$. Permutations apply this principle by reducing the available choices by one for each subsequent position.

• When arranging a subset of $r$ items from a larger pool of $n$ items, the formula is derived by stopping the factorial multiplication after $r$ positions. This leads to the standard notation $^nP_r$.

The Permutation Formula: $^nP_r = \frac{n!}{(n-r)!}$

• This formula effectively 'cancels out' the arrangements of the items that were not chosen, leaving only the ordered arrangements of the $r$ selected items.

Items Together (The 'Stuck' Method)

• When specific items must remain adjacent, treat the group of items as a single super-object. Calculate the permutations of the remaining items plus this one super-object.

• Crucially, you must then multiply by the internal permutations of the items within that group. For example, if 3 people must sit together, they can be arranged in $3!$ ways among themselves while moving as a unit.

Items Separated (The 'Gap' Method)

• To ensure items are never adjacent, first arrange the unrestricted items in a row. This creates 'gaps' between them, including the spaces at the very beginning and the very end.

• If there are $k$ unrestricted items, there are $k+1$ available gaps. Use the $^nP_r$ formula to place the restricted items into these gaps, ensuring no two restricted items can ever touch.

• The primary difference between permutations and combinations is the relevance of order. In permutations, 'ABC' and 'CBA' are two different outcomes; in combinations, they are considered the same group.

• Use permutations when the position confers a specific role (e.g., President vs. Vice President) and combinations when the roles are identical (e.g., two members of a committee).

• Identify the 'Repeaters': Always scan the problem for identical items (like letters in a word). Forgetting to divide by the factorial of repeats is the most common way to lose marks.

• The 'At Least' Trap: If a question asks for 'at least' two items together, it is often easier to calculate the complement. Subtract the cases where 'none are together' from the 'total possible arrangements'.

• Slot Method: For complex constraints, draw physical slots (lines) on your paper. Fill the restricted slots first to see how many options remain for the unrestricted positions.

• Sanity Check: Permutation numbers grow extremely fast. If you are arranging 10 items and get a small number like 100, re-check if you should be using factorials ($10!$ is over 3.6 million).