An ice cream shop needs to prepare at least $n$ buckets of different flavors of ice cream to satisfy the advertisement claim that ``the number of combinations of selecting two scoops of different flavors exceeds 100 types.'' How many ways can a customer select two scoops (which may be the same flavor) from $n$ buckets?
(1) 101
(2) 105
(3) 115
(4) 120
(5) 225

Let $a , b , c$ be real numbers and $0 < b < 1$ such that
$$\begin{aligned} & a = b \cdot c \\ & a + c = b \end{aligned}$$
Given this, which of the following orderings is correct?
A) $a < b < c$
B) $a < c < b$
C) $b < a < c$
D) $c < a < b$
E) $c < b < a$

Let $a$ and $b$ be digits. Given the sets
$$\begin{aligned} & A = \{ 5,6,7,8,9 \} \\ & B = \{ 1,4,5,7 \} \\ & C = \{ a , b \} \end{aligned}$$
If the number of elements in the Cartesian product $(A \cup C) \times (B \cup C)$ is 28, what is the sum $a + b$?
A) 5
B) 6
C) 8
D) 9
E) 11

In a mathematics class, the teacher asks Veli to calculate in how many different ways 3 students can be selected, Yasin to calculate in how many different ways 5 students can be selected, and Zeynep to calculate in how many different ways 11 students can be selected from the students in the class. All three students calculated the requested numbers correctly.
Given that the numbers found by Yasin and Zeynep are the same positive integer, what is the number found by Veli?
A) 364 B) 560 C) 688 D) 816 E) 960

Duru observed that she was present in 45 of all three-person groups that could be formed from the students in her class.
Accordingly, in how many of all three-person groups that could be formed from the students in Duru's class is Duru not present?
A) 20
B) 35
C) 90
D) 105
E) 120