Five balls labeled with the numbers $1,2,3,4,5$ are to be placed into 3 boxes $\mathrm { A } , \mathrm { B } , \mathrm { C }$. How many ways are there to place the balls in the boxes such that no box has a sum of the numbers on the balls that is 13 or more? (Note: For an empty box, the sum of the numbers on the balls is considered to be 0.) [4 points]
(1) 233
(2) 228
(3) 222
(4) 215
(5) 211

At a certain event venue, there are 5 locations where one banner can be installed at each location. There are three types of banners: A, B, and C, with 1 banner of type A, 4 banners of type B, and 2 banners of type C. When selecting and installing 5 banners at the 5 locations to satisfy the following conditions, how many possible outcomes are there? (Note: banners of the same type are not distinguished from each other.) [3 points]
(a) Banner A must be installed.
(b) Banner B is installed in at least 2 locations.
(1) 55
(2) 65
(3) 75
(4) 85
(5) 95

At a certain event venue, there are 5 locations where one banner can be installed at each location. There are three types of banners: A, B, and C, with 1 banner of type A, 4 banners of type B, and 2 banners of type C. When selecting and installing 5 banners at the 5 locations to satisfy the following conditions, how many possible cases are there? (Note: Banners of the same type are not distinguished from each other.) [3 points] (가) Banner A must be installed. (나) Banner B must be installed in at least 2 locations.
(1) 55
(2) 65
(3) 75
(4) 85
(5) 95

In how many ways can 4 bottles of the same type of juice, 2 bottles of the same type of water, and 1 bottle of milk be distributed to 3 people without remainder? (Note: Some people may not receive any bottles.) [3 points]
(1) 330
(2) 315
(3) 300
(4) 285
(5) 270

Four students participate in garbage classification publicity activities in 3 residential areas, with each student going to only 1 area and each area having at least 1 student assigned. The total number of different arrangement methods is $\_\_\_\_$.

From 6 people, select 4 to work on duty, each person works for 1 day. The first day needs 1 person, the second day needs 1 person, the third day needs 2 people. There are $\_\_\_\_$ ways to arrange them.

In the $4 \times 4$ grid table shown in the figure, select 4 squares such that each row and each column contains exactly one selected square. The total number of ways to do this is $\_\_\_\_$. Among all selections satisfying the above requirement, the maximum sum of the 4 numbers in the selected squares is $\_\_\_\_$.

11	21	31	40
12	22	33	42
13	22	33	43
15	24	34	44

Suppose 40 distinguishable balls are to be distributed into 4 different boxes such that each box gets exactly 10 balls. Out of these 40 balls, 10 are defective and 30 are non-defective. In how many ways can the balls be distributed such that all the defective balls go to the first two boxes?
(A) $\frac{40!}{(10!)^4}$
(B) $\frac{30! \cdot 20!}{(10!)^5}$
(C) $\frac{20! \cdot 20!}{(10!)^5}$
(D) $\frac{30! \cdot 10!}{(10!)^4}$

There are six boxes numbered from 1 to 6. We are to put four balls of different sizes into these boxes.
(1) There are altogether $\mathbf{AA}$ ways to put the four balls into the boxes.
(2) There are $\mathbf{CDE}$ ways to put the four balls into four separate boxes.
(3) There are $\mathbf{FGH}$ ways to put three balls into one box and the fourth ball into another.
(4) There are $\mathbf{IJK}$ ways to put at least one ball into the box numbered 1.

All 5 different marbles are to be distributed among 3 siblings of different ages.
In how many different ways can this distribution be made such that the oldest sibling gets 1 marble and the other two each get at least one marble?
A) 45
B) 50
C) 60
D) 70
E) 75