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| Combinatorics -
Combinatorial
Analysis |
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Permutations
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Permutations
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Permutations
of n objects some of which are the
same |
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Combinations
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Combinations
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Combinations
with repetition
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| Permutations
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| Given
a set of n
different elements or objects. Any distinct ordered arrangement of the n
elements is called |
| permutation. |
| The total
number of permutations for n
elements is |
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| Example:
Given is the sequence of four digits 1, 2, 3, 4.
Write all possible ordered arrangements or |
| permutations of the 4
digits. |
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Solution:
The number of
permutations of the given 4 digits, P(4)
= 4! = 4 · 3 · 2 · 1 = 24.
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| The
permutations are, |
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1, 2, 3, 4
2, 1, 3, 4
3, 1, 2, 4
4, 1, 2, 3 |
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1, 2, 4, 3
2, 1, 4, 3
3, 1, 4, 2
4, 1, 3, 2 |
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1, 3, 2, 4
2, 3, 1, 4
3, 2, 1, 4
4, 2, 1, 3 |
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1, 3, 4, 2
2, 3, 4, 1
3, 2, 4, 1
4, 2, 3, 1 |
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1, 4, 2, 3
2, 4, 1, 3
3, 4, 1, 2
4, 3, 1, 2 |
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1, 4, 3, 2
2, 4, 3, 1
3, 4, 2, 1
4, 3, 2, 1. |
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| Permutations
of n objects some of which are the
same |
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The number of
permutations of n
elements some groups of which are the same
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| where,
k1,
k2,
... , km
denotes each group with identical elements. |
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| Example:
How many different
7-letter words can be formed from the word GREETER? |
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Solution:
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| since
the letter R repeats twice and E repeats 3 times. |
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| Example:
How many four-digit
numbers can be written with all of the digits 2, 3, 3, 4 and
write them in |
| increasing order. |
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Solution: In
the given sequence of four digits, the digit 3 repeat twice, so |
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| the
12 four-digit
numbers written in increasing
order are, |
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2 3 3 4 3
2 3 4
4 2 3 3 |
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2 3 4 3 3
2 4 3 4
3 2 3 |
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2 4 3 3 3
3 2 4 4
3 3 2. |
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3
3 4 2 |
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3
4 2 3 |
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3
4 3 2 |
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| Combinations
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| Given
a set of n
different elements or objects. Select a subset of r
elements out of n.
Such selection is |
| called the combination. |
| A
combination is an unordered arrangement of r
objects selected from n
different objects taken r
at a time. |
| The number of
distinct combinations selecting r
elements out of n
is |
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| Therefore,
combinations must differ from each other at least in one element. |
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| Example:
Find the number of
combinations of size 3 that
can be made from digits 1, 2, 3, 4,
5 and
write |
| them out. |
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Solution: Since,
n = 5 and r
= 3 then |
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| The
combinations are,
1 2 3 2
3 4 3
4 5. |
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1 2 4
2 3 5 |
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1 2 5
2 4 5 |
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1 3 4 |
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1 3 5 |
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1 4 5 |
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| Example:
Find the number of
combinations of size 4 that
can be made from letters A, B, C, D, E, F and |
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write them out. |
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Solution: Since,
n = 6 and r
= 4 then |
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| The
combinations are,
A B C D B
C D E C
D E F. |
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A B C E
B C D F |
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A B C F
B C E F |
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A B D E
B D E F |
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A B D F |
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A B E F |
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A C D E |
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A C D F |
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A C E F |
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A D E F |
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| Combinations
with repetition
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| The number of ways to
choose r
objects from a set of n
different objects, so that an object can be chosen |
| more than
once |
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| Remember
that combinations must differ from each other at least in one
element. |
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| Example:
Find the number of
combinations of size 3 that
can be made from digits 1, 2, 3, 4 if repetition is |
| allowed, and
write them out. |
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Solution: Since,
n = 4 and r
= 3 then |
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| The
combinations are,
1 1 1 2
2 2 3
3 3 4
4 4 1 2 3 |
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1 1 2 2
2 1 3
3 1 4
4 1 1 2
4 |
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1 1 3 2
2 3 3
3 2 4
4 2 1 3 4 |
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1 1 4 2
2 4 3
3 4 4
4 3 2 3 4 |
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| Example:
Find the number of
combinations of size 4 that
can be made from letters A, B, C if repetition is |
| allowed, and
write them out. |
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Solution: Since,
n = 3 and r
= 4 then |
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| The
combinations with repetition are,
A A A A B
B B B C
C C C A
A B C |
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A A A B B
B B C C
C C A B
B A C |
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A A B B B
B C C C
C A A C
C A B. |
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A B B B B
C C C C
A A A |
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| Combinatorics
and probability contents |
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