When 6 different balls are placed 3 each in two baskets A and B, how many possible outcomes are there? [3 points]

A group of 64 players in a chess tournament needs to be divided into 32 groups of 2 players each. In how many ways can this be done?
(A) $\frac { 64 ! } { 32 ! 2 ^ { 32 } }$
(B) $\binom { 64 } { 2 } \binom { 62 } { 2 } \cdots \binom { 4 } { 2 } \binom { 2 } { 2 }$
(C) $\frac { 64 ! } { 32 ! 32 ! }$
(D) $\frac { 64 ! } { 2 ^ { 64 } }$

A group of 9 students, $s _ { 1 } , s _ { 2 } , \ldots , s _ { 9 }$, is to be divided to form three teams $X , Y$, and $Z$ of sizes 2,3 , and 4 , respectively. Suppose that $s _ { 1 }$ cannot be selected for the team $X$, and $s _ { 2 }$ cannot be selected for the team $Y$. Then the number of ways to form such teams, is $\_\_\_\_$ .

The set $S = \{ 1,2,3 , \ldots , 12 )$ is to be partitioned into three sets $A , B , C$ of equal size. Thus, $A \cup B \cup C = S , A \cap B = B \cap C = A \cap C = \phi$. The number of ways to partition $S$ is
(1) $\frac { 12 ! } { 3 ! ( 4 ! ) ^ { 3 } }$
(2) $\frac { 12 ! } { 3 ! ( 3 ! ) ^ { 4 } }$
(3) $\frac { 12 ! } { ( 4 ! ) ^ { 3 } }$
(4) $\frac { 12 ! } { ( 3 ! ) ^ { 4 } }$

Eight persons are to be transported from city $A$ to city $B$ in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is
(1) 1120
(2) 3360
(3) 1680
(4) 560

Let the set $S = \{ 2,4,8,16 , \ldots , 512 \}$ be partitioned into 3 sets $A , B , C$ with equal number of elements such that $\mathrm { A } \cup \mathrm { B } \cup \mathrm { C } = \mathrm { S }$ and $\mathrm { A } \cap \mathrm { B } = \mathrm { B } \cap \mathrm { C } = \mathrm { A } \cap \mathrm { C } = \phi$. The maximum number of such possible partitions of $S$ is equal to:
(1) 1680
(2) 1640
(3) 1520
(4) 1710

A graduating class has 8 students responsible for planning a class trip, divided into three groups A, B, and C, consisting of 3, 3, and 2 people respectively. Each of the 8 students will be assigned to one of the groups, and two students, A and B, must be in the same group. How many ways are there to divide the 8 students into groups?
(1) 140 ways
(2) 150 ways
(3) 160 ways
(4) 170 ways
(5) 180 ways

Four distinct marbles will be distributed to 3 siblings such that each sibling receives at least 1 marble.
In how many different ways can this distribution be done?
A) 24
B) 32
C) 36
D) 40
E) 48