Permutations

A permutation is an arrangement of objects in a specific order. The order of arrangement matters in permutations. When selecting and arranging \( r \) objects from \( n \) distinct objects, the number of permutations is denoted by \( {}^nP_r \) or \( P(n, r) \).

Formula Derivation

The number of permutations of \( n \) distinct objects taken \( r \) at a time is given by:

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

where \( n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \) and \( 0! = 1 \).

Special cases:

• When \( r = 0 \), \( {}^nP_0 = 1 \).
• When \( r = n \), \( {}^nP_n = n! \).
• Permutations of \( n \) objects with repetitions allowed: \( n^r \).
• Permutations of \( n \) objects with \( p_1, p_2, \ldots, p_k \) alike objects respectively:

\[ \frac{n!}{p_1! p_2! \cdots p_k!} \]

Worked Illustration

Find the number of ways to arrange 3 objects A, B, and C taking 2 at a time.

Using the formula:

\[ {}^3P_2 = \frac{3!}{(3-2)!} = \frac{3 \times 2 \times 1}{1!} = 6 \]

The arrangements are AB, AC, BA, BC, CA, CB.

Solved Example

Example: How many 3-letter words can be formed from the letters of the word "MATH" without repetition?

Solution:

Number of letters \( n = 4 \), number chosen \( r = 3 \).

Number of permutations:

\[ {}^4P_3 = \frac{4!}{(4-3)!} = \frac{4 \times 3 \times 2 \times 1}{1!} = 24 \]

So, 24 different 3-letter words can be formed.

Practice Set

• Level 1 – Easy: Find \( {}^5P_2 \).
• Level 2 – Moderate: How many ways can 4 students be arranged in a row?
• Level 3 – Challenging: Find the number of permutations of the letters in the word "BALLOON".

Answer Key

• \( {}^5P_2 = \frac{5!}{3!} = 20 \)
• Number of ways to arrange 4 students = \( 4! = 24 \)
• Number of permutations of "BALLOON":
\[ \frac{7!}{1! \times 2! \times 2! \times 1! \times 1!} = \frac{5040}{4} = 1260 \]

Quick Reference

Concept	Formula
Permutation of \( n \) objects taken \( r \) at a time	\( {}^nP_r = \frac{n!}{(n-r)!} \)
Permutation with repetition	\( n^r \)
Permutation of objects with alike items	\( \frac{n!}{p_1! p_2! \cdots p_k!} \)

Glossary

• Permutation: Arrangement of objects where order matters.
• Factorial (\( n! \)): Product of all positive integers up to \( n \).
• Repetition: Allowing objects to be repeated in arrangements.
• Alike objects: Objects that are indistinguishable from each other.

Combinations

A combination is a selection of objects where order does not matter. When choosing \( r \) objects from \( n \) distinct objects without regard to order, the number of combinations is denoted by \( {}^nC_r \) or \( C(n, r) \).

Formula Derivation

The number of combinations of \( n \) distinct objects taken \( r \) at a time is given by:

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

This formula is derived from permutations by dividing the number of permutations by the number of ways to arrange \( r \) objects:

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

Worked Illustration

Find the number of ways to select 2 objects from 4 distinct objects.

Using the formula:

\[ {}^4C_2 = \frac{4!}{2! (4-2)!} = \frac{24}{2 \times 2} = 6 \]

Solved Example

Example: How many committees of 3 members can be formed from 7 people?

Solution:

Number of people \( n = 7 \), committee size \( r = 3 \).

Number of combinations:

\[ {}^7C_3 = \frac{7!}{3! (7-3)!} = \frac{5040}{6 \times 24} = 35 \]

So, 35 different committees can be formed.

Practice Set

• Level 1 – Easy: Calculate \( {}^6C_1 \).
• Level 2 – Moderate: Find the number of ways to choose 4 books from 10.
• Level 3 – Challenging: Prove that \( {}^nC_r = {}^nC_{n-r} \).

Answer Key

• \( {}^6C_1 = 6 \)
• Number of ways to choose 4 books from 10:
\[ {}^{10}C_4 = \frac{10!}{4!6!} = 210 \]
• Proof of \( {}^nC_r = {}^nC_{n-r} \):
\[ {}^nC_r = \frac{n!}{r!(n-r)!} = \frac{n!}{(n-r)!r!} = {}^nC_{n-r} \]

Quick Reference

Concept	Formula
Combination of \( n \) objects taken \( r \) at a time	\( {}^nC_r = \frac{n!}{r! (n-r)!} \)
Relation between permutation and combination	\( {}^nC_r = \frac{{}^nP_r}{r!} \)
Symmetry property	\( {}^nC_r = {}^nC_{n-r} \)

Glossary

• Combination: Selection of objects where order does not matter.
• Factorial (\( n! \)): Product of all positive integers up to \( n \).
• Symmetry property: Number of combinations choosing \( r \) equals choosing \( n-r \).
• Committee: A group selected from a larger set.