Classification of Combinatorial problems

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Classification of Combinatorial problems
i) Order matters / does not matter

• Choose a committee of 3 out of 10 members
• (members are distinguishable) the order of the
  
• members in committee does not matter

C (10, 3)

• Choose president, vice-president and secretary
• out of 10 members three different assignments
  mean
• the order is important

10?9?8
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ii) identical objects

• Count different arrangements of 4 objects AABC

Modify the problem AaBC
Then we have 4!24 distinguishable arrangements
For any positions of B and C we have 2 choices,
that should not be distinguished. So only
24/212 arrangements are distinct

• Count arrangements of AABB

12/26
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iii) with/without repetitions
26?25?24
263

• You draw 3 balls from a box with 10 red, 10 blue
  and 10green

(order is not important rbgbrggbr )

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A committee of 3 is to be chosen from 5
Democrats, 3 Republicans and 4 Independents.
a) In how many ways it can be done?

• all representatives are not distinguished

• all members of a committee are equivalent

• selections with repetition

• the same problem 5 red, 3 blue and 4 green
  balls are
• picked at random. How many different outcomes are
  possible?

DDD IDD DDR IDR DRR IRR RRR IIR I I I IID
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Consider the situation, when all representatives
are distinguished. What will be the number of
possible committees?
C (12, 3)
b) In how many ways it can be done if the
committee must contain at least one independent?
We must put one star in the independent box and
count different ways to distribute two remaining
stars in three boxes.
DDI DRI DII RRI RII III
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For what situations the following answers apply ?
3 distinguished positions in the committee and
all representatives are distinguished
12?11?10
12 distinguished positions in the
committee, representatives of a party are not
distinguished
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In how many ways can we select three books each
from different subject from a set of 6
distinct history books, 9 distinct classic
books, 7 distinct law books, and 4 distinct
education books?
C (6,3) ? C (9,3) ? C (7,3) ? C (4,3)
An exam has 12 problems. How many ways can
integer points be assigned to the problems if the
total of the points is 100 and each problem is
worth at least five points?
x1 x2 x12100 ( xi ?5 )
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Find the formula involving the connectives ?, ?,
and ? that has the following truth table
You can observe that q r ? (r?q)
? ?r?q
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Let ? be an unknown boolean logical operator. The
logical statement (p ? q) ? r ? (q ? r) is
equivalent to ?(p ? q) ? r. Given this
information, there are 2 possible truth tables
for the boolean logical operator ?. List both of
these truth tables.
q ? r 1 0 1 1/0 1 1/0 1 1/0
p q r p?q (p ? q)?r 1 1 1
1 1 1 1 0 1 1 1 0 1
0 1 1 0 0 0 0 0 1 1 0 1 0 1
0 0 0 0 0 1 0 1 0 0 0 0 0
?(p ? q) ? r (p ? q) ? r ? (q ?
r) 1 0 1 1 1 1 1 1
10
Answer
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Use laws of logic to show that the following
expression is a tautology p ? ( p?q ) ? (q ?
r ) ? (? q ? r )
? (q ? p) ? (q ? r )
( p?q ) ? (q ? r )
? q ? (p ? r )
( p?q ) ? (q ? r ) ? (? q ? r ) ? q ? (p ? r
) ? (? q ? r )
? (q ? ? q) ? (p ? r ) ? r
? T ? (p ? r ) ? r
? T
p ? T ?T
12
In each case determine whether or not two
propositions are logically equivalent
?
?
?
?
13
Suppose A ? B and C is any set. Prove or disprove
that C ?B ? C ?A.
Proof.
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x ? C ?B ? x ? C ?A
Contrapositive
x ? C ?A ? x ? C ?B
x ? C ?B
? ? (x ?C ? x?A)
? (x ? C ? x ? A)
? (x ? C ? x ? B)
? ?(x ? C ? x ? B)
? x ? C ?B