A children's toy truck-carrier is formed by a trailer and ten small cars transported on it. In the production sector of the company that manufactures this toy, all the small cars are painted so that the toy looks more attractive. The colors used are yellow, white, orange and green, and each small car is painted with only one color. The truck-carrier has a fixed color. The company determined that in every truck-carrier there must be at least one small car of each of the four available colors. Change of position of the small cars on the truck-carrier does not generate a new model of the toy.
Based on this information, how many distinct models of the truck-carrier toy can this company produce?
(A) $C_{6,4}$
(B) $C_{9,3}$
(C) $C_{10,4}$
(D) $6^{4}$
(E) $4^{6}$

10. In how many ways can 10 identical chocolate bars be distributed among 5 children, in such a way that each child gets at least one chocolate bar?
(a) 50
(b) 126
(c) 252
(d) 3125

A certain volunteer service center operates the following four volunteer activity programs every day.

Program	A	B	C	D
Volunteer Activity Hours	1 hour	2 hours	3 hours	4 hours

Chulsu wants to participate in one program each day for 5 days at this volunteer service center and create a volunteer activity plan so that the total volunteer activity hours is 8 hours. How many different volunteer activity plans can be created? [4 points]
(1) 47
(2) 44
(3) 41
(4) 38
(5) 35

There are 8 white ping-pong balls and 7 orange ping-pong balls to be distributed entirely among 3 students. In how many ways can the balls be distributed so that each student receives at least one white ball and at least one orange ball? [4 points]
(1) 295
(2) 300
(3) 305
(4) 310
(5) 315

When selecting 5 numbers from the digits $1,2,3,4$ with repetition allowed, how many cases are there where the digit 4 appears at most once? [3 points]
(1) 45
(2) 42
(3) 39
(4) 36
(5) 33

How many ordered pairs $( x , y , z , w )$ of non-negative integers satisfy the system of equations $$\left\{ \begin{array} { l } x + y + z + 3 w = 14 \\ x + y + z + w = 10 \end{array} \right.$$ ? [4 points]
(1) 40
(2) 45
(3) 50
(4) 55
(5) 60

Find the number of all ordered pairs $( a , b , c )$ of natural numbers satisfying the following conditions. [4 points] (가) $a \times b \times c$ is odd. (나) $a \leq b \leq c \leq 20$

How many ordered pairs $( a , b , c , d , e )$ of non-negative integers satisfy the following conditions? [4 points] (가) Among $a , b , c , d , e$, the number of 0's is 2. (나) $a + b + c + d + e = 10$
(1) 240
(2) 280
(3) 320
(4) 360
(5) 400

Find the number of all ordered pairs $( a , b , c )$ of non-negative integers satisfying the following conditions. [4 points] (가) $a + b + c = 7$ (나) $2 ^ { a } \times 4 ^ { b }$ is a multiple of 8.

Three students A, B, and C are given 6 identical candies and 5 identical chocolates to be distributed completely according to the following rules. Find the number of ways to do this. [4 points] (가) The number of candies that student A receives is at least 1. (나) The number of chocolates that student B receives is at least 1. (다) The sum of the number of candies and chocolates that student C receives is at least 1.

Find the number of ways to distribute 6 black hats and 6 white hats among four students $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ according to the following rules without remainder. (Note: hats of the same color are not distinguished from each other.) [4 points] (가) Each student receives at least 1 hat. (나) The number of black hats each student receives is different from one another.

How many ordered pairs $( a , b , c , d , e )$ of natural numbers satisfy the following conditions? [3 points]
(a) $a + b + c + d + e = 12$
(b) $\left| a ^ { 2 } - b ^ { 2 } \right| = 5$
(1) 30
(2) 32
(3) 34
(4) 36
(5) 38

Find the total number of ordered quadruples $(a, b, c, d)$ of natural numbers not exceeding 6 that satisfy the following condition. [4 points] $$a \leq c \leq d \text{ and } b \leq c \leq d$$

153. In how many ways can 9 identical books be placed in 5 shelves such that at least one book is placed on each shelf?
(1) $35$ (2) $42$ (3) $56$ (4) $70$

153- How many non-negative integer solutions does the inequality $x + y + z \leq 5$ have?
(1) $50$ (2) $54$ (3) $56$ (4) $60$

147. In how many ways can 11 identical balls be distributed among 5 people such that each person has at least one ball?
(1) $160$ (2) $180$ (3) $210$ (4) $220$
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ریاضیات 121-A صفحه ۸

143- The number of terms in the expansion of $(a+b+c)^{12}$ is:
\[ (1)\quad 72 \qquad (2)\quad 78 \qquad (3)\quad 84 \qquad (4)\quad 91 \]

How many natural numbers less than $10^{8}$ are there, whose sum of digits equals 7?

Let $f(n)$ be the number of ways to write a positive integer as an ordered sum of three non-negative integers, where each integer is chosen from $\{0, 1, 2, \ldots, 2n-1\}$ (i.e., using $n$ colours with values $0$ to $2n-1$). Find $f(n)$.

In how many ways can 20 identical chocolates be distributed among 8 students such that each student gets at least one chocolate and exactly 2 students get at least 2 chocolates?
(A) 308 (B) 280 (C) 300 (D) 320

Find the number of ordered triples $(a, b, c)$ of positive integers such that $abc = 1000$.
(A) 90 (B) 100 (C) 110 (D) 120

Let $n_1 < n_2 < n_3 < n_4 < n_5$ be positive integers such that $n_1 + n_2 + n_3 + n_4 + n_5 = 20$. Then the number of such distinct arrangements $(n_1, n_2, n_3, n_4, n_5)$ is

This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is ${}^{9}\mathrm{C}_{3}$. Statement-2: The number of ways of choosing any 3 places from 9 different places is ${}^{9}\mathrm{C}_{3}$.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is false.
(3) Statement-1 is false, Statement-2 is true.
(4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at least 2 apples, is
(1) 406
(2) 130
(3) 142
(4) 136

Number of ways to distribute 6 identical oranges among 4 persons such that each gets at least one orange is (A) 8 (B) 12 (C) 10 (D) 13