Permutation

Permutation refers to the arrangement of a set of objects in a specific order. When dealing with permutations, the order of the objects is important.

Formula for Permutations

The number of permutations of ( n ) distinct objects taken ( r ) at a time is given by:
[ P(n, r) = \frac{n!}{(n – r)!} ]
where ( n! ) (n factorial) is the product of all positive integers up to ( n ).

Example of Permutation

Suppose we want to find the number of ways to arrange 3 out of 5 distinct letters: ( A, B, C, D, E ).

Using the formula:
[ P(5, 3) = \frac{5!}{(5 – 3)!} = \frac{5!}{2!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} = 5 \times 4 \times 3 = 60 ]

So, there are 60 different ways to arrange 3 letters out of 5.

Detailed Arrangement:

Some of these permutations include:

• ABC
• ACB
• BAC
• BCA
• CAB
• CBA
• and 54 more…

Combination

Combination refers to the selection of objects from a set where the order of the objects does not matter.

Formula for Combinations

The number of combinations of ( n ) distinct objects taken ( r ) at a time is given by:
[ C(n, r) = \binom{n}{r} = \frac{n!}{r!(n – r)!} ]

Example of Combination

Suppose we want to find the number of ways to choose 3 out of 5 distinct letters: ( A, B, C, D, E ).

Using the formula:
[ C(5, 3) = \binom{5}{3} = \frac{5!}{3!(5 – 3)!} = \frac{5!}{3! \cdot 2!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \cdot 2 \times 1} = \frac{120}{6 \times 2} = 10 ]

So, there are 10 different ways to choose 3 letters out of 5 without regard to order.

Detailed Selection:

Some of these combinations include:

• ABC
• ABD
• ABE
• ACD
• ACE
• ADE
• BCD
• BCE
• BDE
• CDE

Notice that in combinations, ( ABC ) is considered the same as ( ACB ), ( BAC ), etc.

Summary

• Permutation: Order matters.
• Example: Arranging 3 out of 5 letters ( A, B, C, D, E ).
• Formula: P(n, r) = (n!)/(n – r)
• Combination: Order does not matter.
• Example: Choosing 3 out of 5 letters ( A, B, C, D, E ).
• Formula: ( C(n, r) = {n}/{r} = (n!)/r!(n – r)!