Permutation and Combination

🔷 Permutation

A permutation is a specific arrangement of a set of objects in a particular order. It refers to the different ways in which a group of items can be organized when the order of the items is important. In other words, changing the sequence or position of the items leads to a different permutation.

For example, arranging the letters A, B, and C as "ABC" is not the same as "CAB"—because the order has changed.

Permutations are commonly used in problems involving ranking, scheduling, assigning positions, or arranging people or things where the position or sequence matters.

🔑 Key Points

Order matters in permutations.
If you have n objects and want to arrange r of them, use the permutation formula:
$P (n, r) = \frac{n!}{(n - r)!}$
n! (n factorial) means:
$n! = n \times (n - 1) \times (n - 2) \times \dots \times 1$

✏️ Example 1: Permutation of All 3 Objects

You have 3 letters: A, B, C.
How many ways can you arrange them?

$P (3, 3) = 3! = 3 \times 2 \times 1 = 6$

Arrangements:

✏️ Example 2: Permutation of 3 Objects Taken 2 at a Time

You want to arrange only 2 of the letters A, B, and C.

$P (3, 2) = \frac{3!}{(3 - 2)!} = \frac{3 \times 2 \times 1}{1!} = \frac{6}{1} = 6$

Arrangements:

📝 Summary of Permutations

Use when order matters.
Formula:
$P (n, r) = \frac{n!}{(n - r)!}$
Example: Arranging letters A, B, C in different orders.

🔷 Combination

A combination is a selection of items from a larger set without considering the order of the items. That means combinations focus only on which items are selected, not the sequence in which they appear.

For example, choosing 3 fruits from a basket of 5 fruits (apple, banana, cherry, date, and elderberry) is a combination problem, because selecting "apple, banana, cherry" is the same as selecting "cherry, apple, banana"—the order does not change the selection.

Combinations are widely used in scenarios involving group selections, committees, teams, menus, or lotteries where the arrangement is irrelevant.

📌 When to Use Combinations

When you're selecting a group or subset of items, and order doesn't matter.
Examples: Choosing team members, selecting items from a menu, picking lottery numbers.

📐 Mathematical Definition

If you have n items and want to choose r of them, the number of combinations is:

$C (n, r) = \frac{n!}{r! (n - r)!}$

Where:

n! = factorial of total items
r! = factorial of selected items
(n - r)! = factorial of unselected items

⚙️ Explanation of the Formula

n! counts all possible arrangements of n items.
Since order doesn't matter:
- Divide by r! to remove order among chosen items.
- Divide by (n - r)! to remove order among unchosen items.

✏️ Example 1: Choosing Fruits

Question: How many ways can you choose 3 fruits from 5: apple, banana, cherry, date, elderberry?

Solution:

$C (5, 3) = \frac{5!}{3! (5 - 3)!} = \frac{5 \times 4 \times 3!}{3! \times 2!} = \frac{5 \times 4}{2 \times 1} = 10$

✅ Answer: 10 ways

✏️ Example 2: Forming a Committee

Question: How many ways can you form a committee of 4 people from a group of 10?

Solution:

$C (10, 4) = \frac{10!}{4! (10 - 4)!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210$

✅ Answer: 210 committees can be formed.

🔄 Comparison Table: Permutations vs. Combinations

Feature	Permutation	Combination
Order	Matters	Doesn’t matter
Used For	Arranging people, digits, alphabets, etc.	Selecting groups, teams, menu items, etc.
Example 1	Choosing a captain, pitcher, and catcher	Choosing 3 players from a team
Example 2	Picking 1st, 2nd, 3rd place winners	Picking any 3 winners (no rank/order)
Formula	$P(n, r) = \frac{n!}{(n - r)!}$	$C(n, r) = \frac{n!}{r!(n - r)!}$

📌 Important Notes

Combinations are always less than or equal to permutations.
Special cases:
- $C(n, 0) = 1 \] (Only one way to choose nothing)$
- $C(n, n) = 1 \] (Only one way to choose everything)$

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