Permutation

PERMUTATION

Chapter 8 of the sources, titled “Permutation”, covers the various ways to calculate ordered arrangements of objects. It progresses from basic definitions to specific formulas for scenarios involving repetition, identical objects, and circular seating.

1. Definition of Permutation

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Ordered Arrangement

A permutation is defined as an ordered arrangement of all or some of $n$ objects. The key factor in permutations is that order matters .

Example: For the set $\{A, B, C\}$ , the arrangements $AB$ and $BA$ are considered distinct permutations.

2. Permutation Formula (Repetition Not Allowed)

When you have $n$ distinct objects and want to arrange $r$ of them without using any object more than once, the formula is:

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Formula ( $nP_r$ )

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

Special Cases:
- $nP_0 = 1$
- $nP_1 = n$
- $nP_n = n!$

3. Permutation with Repetition

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Formula ( $n^r$ )

If you are arranging $r$ objects from a collection of $n$ objects and you are allowed to use the same object multiple times, the total number of arrangements is: $n \times n \times ... \times n = n^r$

Example: Arranging 2 letters from $\{A, B, C\}$ with repetition allows for $3^2 = 9$ ways.

4. Rearranging Letters (Identical Objects)

When some objects are identical, the number of unique permutations decreases because swapping two identical items doesn’t create a new arrangement.

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Identical Objects Formula

If there are several groups of identical items ( $p_1, p_2, \dots$ ), the formula is: $\frac{n!}{p_1! p_2! \dots p_k!}$

Example: In the word “STATISTICS” (10 letters; S=3, T=3, I=2), the total arrangements are $\frac{10!}{3! 3! 1! 2! 1!} = 50,400$ .

5. Circular Permutation

Arranging objects in a circle is different from arranging them in a line because there is no fixed starting point.

Clockwise and anticlockwise are different: $(n - 1)!$ .
Clockwise and anticlockwise are the same: $\frac{(n - 1)!}{2}$ .

Practice Session

Selecting Leadership

From a committee of 8 persons, in how many ways can we choose a chairman and a vice-chairman, assuming one person cannot hold more than one position?

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Here $n = 8$ and $r = 2$ . Using the formula $nP_r$ : $8P_2 = \frac{8!}{(8 - 2)!} = \frac{8!}{6!} = 8 \times 7 = \mathbf{56 \text{ ways}}$

Rearranging Words

How many ways can the letters in the word “DATA” be rearranged?

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Total letters $n = 4$ . The letter ‘A’ repeats $p = 2$ times. $\text{Ways} = \frac{4!}{2!} = \frac{24}{2} = \mathbf{12 \text{ ways}}$

Solving for $n$

Find the value of $n$ if $nP_4 = 20 \times nP_2$ .

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Expand the formula: $\frac{n!}{(n - 4)!} = 20 \times \frac{n!}{(n - 2)!}$ .
Simplify: $\frac{1}{(n - 4)!} = \frac{20}{(n - 2)(n - 3)(n - 4)!}$ .
$(n - 2)(n - 3) = 20 \implies n^2 - 5n + 6 = 20$ .
$n^2 - 5n - 14 = 0 \implies (n - 7)(n + 2) = 0$ .
Since $n$ must be positive, $n = 7$ .

Round Table Seating

Seven people are going to sit at a round table. How many different ways can this be done?

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Using the circular permutation formula $(n - 1)!$ : $(7 - 1)! = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = \mathbf{720 \text{ ways}}$

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The Musical Chairs Analogy

Think of linear permutations like people sitting in numbered theatre seats; if the first person moves to the end, everyone looks different relative to their seat number.

Circular permutations are like a game of musical chairs where everyone moves one seat to the left—the people sitting next to you haven’t changed, so the “arrangement” remains the same until someone actually changes their relative position.