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Counting - Vendiagram

For COMPETITION
Number of Total Problems: 2.
FOR PRINT ::: (Book)

Problem Num : 1

From : NCTM

Type: None

Section:Counting

Theme:None
Adjustment# :

Difficulty: 1

Category Vendiagram

Analysis

Solution/Answer

Problem Num : 2

From : AMC10B

Type:

Section:Counting

Theme:
Adjustment# : 0

Difficulty: 1

'
How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$ ?
$mathrm{(A)} 501 qquad mathrm{(B)} 668 qquad mathrm{(C)} 835 qquad mathrm{(D)} 1002 qquad mathrm{(E)} 1169$
'

Category Vendiagram

Analysis

Solution/Answer
We can use the Principle of Inclusion-Exclusion to solve the problem as follows: We can count the number of multiples of $3$ that are less than $2005$ , add the number of multiples of $4$ that are less than $2005$ , and subtract the number of multiples of $12$ twice that are less than $2005$ (since those are counted twice in each of the $3$ and $4$ cases). Calculating, we get $leftlfloordfrac{2005}{3} ight floor+leftlfloordfrac{2005}{4} ight floor-2*leftlfloordfrac{2005}{12} ight floor...$ (where $lfloor x floor$ denotes the floor function).

Answer:

Array ( [0] => 3729 [1] => 8037 ) 2