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

Five balls labeled with the numbers $1,2,3,4,5$ are to be placed into 3 boxes A, B, C. In how many ways can the balls be placed so that no box contains balls whose numbers sum to 13 or more? (Here, for an empty box, the sum of the numbers on the balls is taken as 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