Chapter 6: Permutations & Combinations

1. Fundamental Principle of Counting

If an event can occur in \( m \) different ways, and another event can occur in \( n \) different ways, then:

Principle	Operation	Keyword
Multiplication	\( m \times n \)	AND (Both happen)
Addition	\( m + n \)	OR (Either happens)

2. Factorials (n!)

The factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \).

\( n! = n \times (n-1) \times (n-2) \times ... \times 1 \). Note: \( 0! = 1 \).

Factorial Factory

Compute \( n! \) and see the expansion.

3. Permutations (Arrangement)

A permutation is an arrangement in a definite order of a number of objects taken some or all at a time.

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

Restricted Permutations

Number of permutations of \( n \) objects where \( p \) are alike: \( \frac{n!}{p!} \)
Permutations with \( p \) alike of one kind, \( q \) alike of another: \( \frac{n!}{p!q!} \)

Word Scrambler

Calculate arrangements of letters in a word.

3.1 Permutations of Objects with Repetition

If among n objects:

p objects are alike of one kind
q objects are alike of another kind
r objects are alike of another kind

Number of permutations:

\( \frac{n!}{p!q!r!} \)

NCERT Example: Number of arrangements of letters of the word MATHEMATICS.

Exam Tip: Count repetitions carefully before applying the formula.

3.2 Circular Permutations

Arrangements of objects in a circle where there is no fixed starting point.

General Case: Arranging \( n \) distinct objects in a circle: \( (n-1)! \)
Necklace/Garland Case: When clockwise and anti-clockwise arrangements are not distinguishable (e.g., beads): \( \frac{(n-1)!}{2} \)

3.3 Derangements

A derangement is a permutation in which none of the objects appear in their original positions.

Formula for \( n \) objects:

\( D_n = n! \left( 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + (-1)^n \frac{1}{n!} \right) \)

Example: Number of ways 3 letters can be placed into 3 corresponding envelopes such that no letter goes into its correct envelope is \( D_3 = 3! \left(1 - 1 + \frac{1}{2} - \frac{1}{6}\right) = 2 \).

4. Combinations (Selection)

Combination is a selection of items from a collection, such that the order of selection does not matter.

Formula: \( ^nC_r = \frac{n!}{r!(n-r)!} \)

Property: \( ^nC_r = ^nC_{n-r} \)

P vs C Showdown

Compare Permutations (Order) vs Combinations (Groups).

Important Properties of Combinations

\( ^nC_0 = ^nC_n = 1 \)
\( ^nC_1 = n \)
\( ^nC_r = ^nC_{n-r} \)
\( ^nP_r = r! \times ^nC_r \)

Exam Tip: Use identities to simplify instead of direct calculation.

4.1 Selection with Conditions

When conditions like at least, at most, or exactly are given:

Break the problem into cases
Solve each case separately
Add the results

Example:

Select 3 students from 5 boys and 4 girls, with at least 1 girl.

Case 1: 1 girl + 2 boys

Case 2: 2 girls + 1 boy

Case 3: 3 girls

Exam Tip: Never use one formula blindly.

Common Mistakes to Avoid

Using permutations when order does not matter
Forgetting to divide by repeated factorials
Applying \( ^nC_r \) when r > n
Missing hidden conditions in word problems

One-Page Revision Checklist

Fundamental Counting Principle

✔ Understand AND → Multiply
✔ Understand OR → Add
✔ Use brackets when steps depend on each other

Factorials

✔ \( n! = n \times (n-1)! \)
✔ \( 0! = 1 \)
✔ Factorials cancel — simplify before calculating

Permutations (Order Matters)

✔ Formula: \( ^nP_r = \dfrac{n!}{(n-r)!} \)
✔ Used for arrangements, rankings, seating
✔ Repetition case: \( \dfrac{n!}{p!q!r!} \)

Combinations (Order Does NOT Matter)

✔ Formula: \( ^nC_r = \dfrac{n!}{r!(n-r)!} \)
✔ Used for selection, groups, teams
✔ Property: \( ^nC_r = ^nC_{n-r} \)

Permutation vs Combination

✔ Ask first: Does order matter?
✔ YES → Permutation
✔ NO → Combination

Problems with Conditions

✔ Keywords: at least, at most, exactly
✔ Break into cases
✔ Solve each case separately
✔ Add results

Common Exam Mistakes

Using permutations instead of combinations
Forgetting repeated letters factorial
Applying \( ^nC_r \) when \( r > n \)
Missing hidden conditions in word problems

CBSE Exam Focus

✔ Word problems are guaranteed
✔ 3–4 mark HOTS questions common
✔ Clear method marks even if answer wrong

Self-Check (Answer Without Looking)

Why is \( 0! = 1 \)?
Difference between \( ^5P_2 \) and \( ^5C_2 \)?
Arrangements of “BANANA”?
Selecting 3 students with at least 1 girl?

Concept Mastery Quiz

1. Value of \( 0! \) is:

2. In permutations, order of arrangement:

3. Formula for \( ^nC_r \) is:

4. How many ways to select 2 players from 5?

5. \( ^nC_n \) is equal to:

Permutations & Combinations