Display Problems

Imagination is more important than knowledge... Albert Einstein

Guess is more important than calculation --- Knowhowacademy.com

Counting - Combinations

SOURCE:COMPETITION
Number of Problems: 5.
FOR PRINT ::: (Book)

Problem Num : 1

Type: Complex

Topic:Counting

Theme:Euqivalence
Adjustment# :

Difficulty: 3

Category Combinations

Analysis

Solution/Hint

Problem Num : 2

Type: None

Topic:Counting

Theme:None
Adjustment# :

Difficulty: 3

Category Combinations

Analysis

Solution/Hint

Problem Num : 3

Type: None

Topic:Counting

Theme:None
Adjustment# :

Difficulty: 1

Category Combinations

Analysis

Solution/Hint

Problem Num : 4

Type:

Topic:Counting

Adjustment# : 0

Difficulty: 1

'
Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?
$mathrm{(A)} 8qquadmathrm{(B)} 9qquadmathrm{(C)} 10qquadmathrm{(D)} 12qquadmathrm{(E)} 16$
'

Category Combinations

Analysis

Solution/Hint
For any given pair of circles, they can intersect at most $2$ times. Since there are ${4choose 2} = 6$ pairs of circles, the maximum number of possible intersections is $6 cdot 2 = 12$ . We can construct such a situation as below, so the answer is $oxed{mathrm{(D)} 12}$ .

Problem Num : 5

Type:

Topic:Counting

Adjustment# : 0

Difficulty: 1

'
In the BIG N, a middle school football conference, each team plays every other team exactly once. If a total of 21 conference games were played during the 2012 season, how many teams were members of the BIG N conference?
$extbf{(A)}hspace{.05in}6qquad extbf{(B)}hspace{.05in}7qquad extbf{(C)}hspace{.05in}8qquad extbf{(D)}hspace{.05in}...$
'

Category Combinations

Analysis

Solution/Hint
This problem is very similar to a handshake problem. We use the formula $frac{n(n-1)}{2}$ to usually find the number of games played (or handshakes). Now we have to use the formula in reverse.
So we have the equation $frac{n(n-1)}{2} = 21$ . Solving, we find that the number of teams in the BIG N conference is $oxed{ extbf{(B)} 7}$ .