Permutations
Permutations: Permutations refer to the arrangement of objects in a
specific order. The order of arrangement is important in permutations.
Definition: A permutation of a set of objects is an arrangement of
those objects in a specific sequence.
Formula for Permutations: If we have a set of nnn objects and
we want to arrange rrr of them, the number of permutations is
given by:
where n! (n factorial) is the product of all positive integers up to n.
Example: Suppose we have 4 objects: A, B, C, and D. How many
ways can we arrange 3 out of these 4 objects?
Therefore, there are 24 different ways to arrange 3 out of 4 objects.
To visualize:
ABC, ABD, ACB, ACD, ADB, ADC
BAC, BAD, BCA, BCD, BDA, BDC
CAB, CAD, CBA, CBD, CDA, CDB
DAB, DAC, DBA, DBC, DCA, DCB
Combinations
Combinations: Combinations refer to the selection of objects without
regard to the order of arrangement. The order does not matter in
combinations.
Definition: A combination of a set of objects is a selection of
those objects without considering the order.
Formula for Combinations: If we have a set of n objects and we
want to select r of them, the number of combinations is given by:
Example: Suppose we have 5 objects: A, B, C, D, and E. How
many ways can we select 2 out of these 5 objects?
Therefore, there are 10 different ways to select 2 out of 5 objects.
To visualize:
AB, AC, AD, AE, BC, BD, BE, CD, CE, DE
Example 2: Arranging Letters How many ways can we arrange the
letters in the word "DOG"?
P(3,3)=3!=6
Therefore, there are 6 different ways to arrange the letters:
DOG, DGO, ODG, OGD, GDO, GOD
Key Differences:
1. Order Matters: In permutations, the order of arrangement is
important, while in combinations, the order does not matter.
2. Formula Differences: The formulas for permutations and
combinations differ due to the consideration of order.
Applications:
1. Permutations: Used in situations where the order of arrangement
is important, such as seating arrangements, arranging books on a
shelf, or assigning tasks.
2. Combinations: Used in situations where the order does not matter,
such as selecting a committee, choosing a subset of items, or
lottery drawings.
Practice Problems:
1. How many ways can you arrange 5 books on a shelf?
P(5,5)=5!=120
Therefore, there are 120 ways to arrange 5 books on a shelf.
2. How many ways can you select 4 cards from a deck of 52 cards?
This calculation is more complex but can be simplified using a
calculator.
3. How many ways can you form a committee of 3 members from a
group of 10 people?
4. In how many ways can you arrange 4 out of 7 different letters?
Therefore, there are 840 ways to arrange 4 out of 7 letters.
Conclusion: Understanding permutations and combinations is crucial
for solving many problems in discrete mathematics and other fields that
involve counting and arranging objects. Practice these concepts to
enhance your problem-solving skills.
