Below is a clear, simple, and exam-oriented explanation of Permutations and Combinations — perfect for BCA/MCA/engineering/discrete structures exams.

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⭐ PERMUTATIONS AND COMBINATIONS

Permutations and combinations are fundamental tools of counting and combinatorics.
They help us count the number of arrangements and selections from a given set of objects.

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⭐ 1. Permutations

A permutation is an arrangement of objects in a specific order.

✔ Order matters in permutations.

Example:
ABC and ACB are different permutations.

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⭐ 1️⃣ Number of Permutations of n Distinct Objects

[
P(n) = n! = n \times (n-1) \times (n-2) \times \ldots \times 1
]

Example:
Permutations of 4 letters = 4! = 24.

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⭐ 2️⃣ Permutations of n Objects Taken r at a Time

Number of arrangements of r items selected from n distinct items:

[
{}^nP_r = \frac{n!}{(n-r)!}
]

✔ Example

How many ways to arrange 3 out of 5 books?

[
{}^5P_3 = \frac{5!}{2!} = \frac{120}{2} = 60
]

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⭐ 3️⃣ Permutations with Repetition Allowed

Each position has n choices, so:

[
n^r
]

Example:
How many 4-digit PIN codes?
Digits = 10
PIN length = 4
→ (10^4 = 10000)

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⭐ 4️⃣ Permutations of Objects with Repetition

If we have objects:

• n total
• where a₁, a₂, …, aₖ are identical repeats

Number of distinct permutations:

[
\frac{n!}{a_1! , a_2! , \cdots , a_k!}
]

Example:
WORD = “BALLOON”

Letters:

• A(1), B(1), L(2), O(2), N(1)

[
\frac{7!}{2!2!} = \frac{5040}{4} = 1260
]

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⭐ COMBINATIONS

A combination is a selection of objects where order does NOT matter.

Example:
AB and BA are same combination.

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⭐ 1️⃣ Combinations of n Objects Taken r at a Time

[
{}^nC_r = \frac{n!}{r!(n-r)!}
]

✔ Example

From 6 students, how many committees of 2 can be made?

[
{}^6C_2 = \frac{6!}{2!4!} = 15
]

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⭐ 2️⃣ Relationship between Permutations and Combinations

[
{}^nP_r = {}^nC_r \cdot r!
]

Reason:
First choose r items (combination) then arrange them (permutation).

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⭐ 3️⃣ Combinations with Repetition (Multisets)

Number of ways to choose r items from n types with repetition allowed:

[
{}^nC_r^{\text{with rep}} = \binom{n+r-1}{r}
]

Example:
Choose 3 candies with 5 flavors:

[
\binom{5+3-1}{3} = \binom{7}{3} = 35
]

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⭐ 4️⃣ Practical Difference

Feature	Permutations	Combinations
Order matters	✔ Yes	✘ No
Formula	(n!/(n-r)!)	(n!/[r!(n-r)!])
Examples	Arranging seats, rankings	Teams, committees

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⭐ Applications

Permutations & combinations are used in:

✔ Probability
✔ Cryptography
✔ Data structures
✔ Computer algorithms
✔ Scheduling
✔ Passwords and PIN generation
✔ Exam question setting
✔ Lottery and games
✔ Counting without listing
✔ Database queries

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⭐ Examples for Practice

1. How many 3-letter words from ABCDE?
   → (5P3 = \frac{5!}{2!} = 60)
2. How many ways to pick 3 fruits from 5?
   → (5C3 = 10)
3. How many 4-digit numbers using digits 1–9 without repetition?
   → (9P4 = 3024)
4. How many ways to arrange the letters of “MISSISSIPPI”?
   → ( \frac{11!}{4!4!2!} = 34650 )

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⭐ Quick Exam-Oriented Summary

Permutation = Arrangement (order matters)

[
{}^nP_r = \frac{n!}{(n-r)!}
]

Combination = Selection (order does not matter)

[
{}^nC_r = \frac{n!}{r!(n-r)!}
]

Relationship

[
{}^nP_r = {}^nC_r \cdot r!
]

With repetition

• Permutations: (n^r)
• Combinations: (\binom{n+r-1}{r})

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