How many non-congruent triangles are there with integer lengths $a \leq b \leq c$ such that $a + b + c = 20$?

When selecting two different odd numbers from the odd numbers from 1 to 30, how many cases are there where the sum of the two numbers is a multiple of 3? [4 points]
(1) 43
(2) 41
(3) 39
(4) 37
(5) 35

For three integers $a , b , c$ satisfying $$1 \leq | a | \leq | b | \leq | c | \leq 5$$ what is the number of all ordered pairs $( a , b , c )$? [4 points]
(1) 360
(2) 320
(3) 280
(4) 240
(5) 200

From 6 cards labeled $1,2,3,4,5,6$ respectively, 2 cards are randomly drawn without replacement. The probability that one of the drawn numbers is a multiple of the other is
A. $\frac { 1 } { 5 }$
B. $\frac { 1 } { 3 }$
C. $\frac { 2 } { 5 }$
D. $\frac { 2 } { 3 }$

Let $S = \{ 1,2,3,4,5,6,9 \}$. Then the number of elements in the set $T = \{ A \subseteq S : A \neq \phi$ and the sum of all the elements of $A$ is not a multiple of $3 \}$ is

Let $x$ and $y$ be distinct integers where $1 \leq x \leq 25$ and $1 \leq y \leq 25$. Then, the number of ways of choosing $x$ and $y$, such that $x + y$ is divisible by 5 , is $\_\_\_\_$ .

Let $S = \{ 1,2,3,5,7,10,11 \}$. The number of non-empty subsets of $S$ that have the sum of all elements a multiple of 3 , is $\_\_\_\_$ .

From the nine numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, any three distinct numbers are randomly selected, with each number having equal probability of being selected. The probability that the product of the three numbers is a perfect square is (20). (Express as a fraction in lowest terms)

In a course, the weekly lesson durations of 7 lessons, each with different lesson times, are given in the table below.

Lesson	Duration (hours)
Lesson 1	5
Lesson 2	4
Lesson 3	4
Lesson 4	5
Lesson 5	3
Lesson 6	5
Lesson 7	5

Aslı, who enrolled in this course, wants to take four different lessons such that the total weekly lesson duration is 17 hours.
Accordingly, in how many different ways can Aslı select the lessons she will take?
A) 8 B) 10 C) 12 D) 16 E) 18