U43 Permutation and Combination (P&C) Essential Formulas

This section covers the fundamental concepts of counting, including the Multiplication Rule, Factorial Notation, Permutations (both linear and circular), and Combinations. Mastery of these principles is crucial for solving a wide range of DSE problems involving selection and arrangement.

1 Fundamental Counting Principles

Multiplication Rule (Fundamental Principle of Counting)

If one operation can be performed in $m$ ways and a second operation can be performed in $n$ ways, then the two operations can be performed together in $m \times n$ ways. This extends to more than two operations.

$$ \text{Total Ways} = m \times n $$
m waysn ways

2 Factorial Notation

Definition of n Factorial

For a positive integer $n$, the factorial of $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$. By convention, $0! = 1$.

$$ n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1 $$

3 Permutations

Permutations of n Distinct Objects (Linear)

The number of ways to arrange $n$ distinct objects in a line is $n!$.

$$ P^n_n = n! $$

Permutations of n Distinct Objects Taken r at a Time

The number of ways to arrange $r$ objects selected from $n$ distinct objects, where order matters.

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

Permutations with Repetition

The number of distinct permutations of $n$ objects where there are $n_1$ identical objects of type 1, $n_2$ identical objects of type 2, ..., $n_k$ identical objects of type $k$.

$$ \frac{n!}{n_1! \times n_2! \times \dots \times n_k!} $$

Circular Permutations

The number of ways to arrange $n$ distinct objects around a circle. Two arrangements are considered the same if one can be rotated to obtain the other.

$$ (n-1)! $$

4 Combinations

Combinations of n Distinct Objects Taken r at a Time

The number of ways to choose $r$ objects from $n$ distinct objects, where order does NOT matter. This is also known as the binomial coefficient.

$$ ^nC_r = C_r^n = \binom{n}{r} = \frac{n!}{r!(n-r)!} $$

Key Properties of Combinations

Useful identities for simplifying calculations.

$$ \binom{n}{r} = \binom{n}{n-r} $$ $$ \binom{n}{0} = \binom{n}{n} = 1 $$ $$ \binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r} \quad \text{(Pascal's Rule)} $$

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