Permutations and Combinations

Learning Objectives

• Understand the Fundamental Principle of Counting and apply it to solve problems involving multiple events occurring in sequence.
• Calculate permutations of distinct objects using the formula $nP_r = \frac{n!}{(n-r)!}$ where repetition is not allowed.
• Determine permutations with repetition using the formula $n^r$ for arranging $n$ objects taken $r$ at a time.
• Solve problems involving permutations of non-distinct objects using the formula $\frac{n!}{p_1! \times p_2! \times \ldots \times p_k!}$.
• Apply the formula for combinations $nC_r = \frac{n!}{r!(n-r)!}$ to find the number of ways to select $r$ objects from $n$ distinct objects.
• Master the use of Factorial Notation where $n!$ represents the product of the first $n$ natural numbers.
• Explore theorems related to permutations and combinations, including relationships and specific cases involving repetition.
• Tackle Restricted Permutations and Arrangements by solving problems with conditions such as specific positions or gaps between objects.
• Address Number Formation and Code Problems by forming numbers or codes with given restrictions on digits or letters.
• Solve Selection Problems with Conditions involving constraints like compulsory items or specific group compositions.

Chapter Notes

Fundamental Principle of Counting

• Definition: If an event can occur in $m$ different ways, following which another event can occur in $n$ different ways, then the total number of occurrences of the events in the given order is $m \times n$.
• Generalization: For any finite number of events, if the first event can occur in $m$ ways, the second in $n$ ways, and the third in $p$ ways, then the total number of occurrences is $m \times n \times p$.

Permutations of Distinct Objects

• Formula: The number of permutations of $n$ different objects taken $r$ at a time is given by: $nP_r = \frac{n!}{(n-r)!}$
• Example: To find the number of 4-letter words from the word 'ROSE', use $4P_4 = 4! = 24$.

Permutations with Repetition

• Formula: The number of permutations of $n$ different objects taken $r$ at a time, where repetition is allowed, is $n^r$.
• Example: If repetition is allowed for the word 'ROSE', the number of 4-letter words is $4^4 = 256$.

Permutations of Non-Distinct Objects

• Formula: The number of permutations of $n$ objects where $p_1$ objects are of one kind, $p_2$ of another, etc., is: $\frac{n!}{p_1! \cdot p_2! \cdot \ldots \cdot p_k!}$
• Example: For the word 'ROOT', with 2 O's, the permutations are $\frac{4!}{2!} = 12$.

Combinations

• Formula: The number of combinations of $n$ different objects taken $r$ at a time is given by: $nC_r = \frac{n!}{r!(n-r)!}$
• Example: For a team of 2 players from X, Y, Z, the combinations are $3C_2 = 3$.

Factorial Notation

• Definition: $n!$ represents the product of the first $n$ natural numbers: $n! = n \times (n-1) \times \ldots \times 2 \times 1$
• Special Case: $0! = 1$

Permutations and Combinations Theorems

• Theorem: $nP_r = nC_r \times r!$
• Application: Useful for converting between permutations and combinations.

Restricted Permutations and Arrangements

• Example: Arranging the word 'DAUGHTER' such that all vowels are together: Treat vowels as a single unit, then arrange.

Number Formation and Code Problems

• Example: Forming numbers greater than 1000000 using digits 1, 2, 0, 2, 4, 2, 4.

Selection Problems with Conditions

• Example: Selecting a committee of 3 from 2 men and 3 women with specific conditions.
• Exercise Problems: Various problems involving permutations and combinations, such as forming numbers, arranging letters, and selecting committees.