144- If $A = \{x \in \mathbb{N},\ 5 < x^2 < 50\}$ and $B = \{3k-2 \mid k \in \mathbb{Z},\ 1 \leq k \leq 4\}$, then the number of elements of $(A \times B) \cap (B \times A)$ is:
(1) $4$ (2) $8$ (3) $16$ (4) $32$

145- The number of subsets of the set $A = \{a, b, c, d, e\}$ that contain exactly one element is:
(1) $10$ (2) $12$ (3) $15$ (4) $20$

\fbox{Workspace}
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146. How many subsets does the set $A = \{1, 2, 3, 4, 5, 6\}$ have that contain exactly two elements?
(1) $8$ (2) $10$ (3) $12$ (4) $15$

146-- In how many ways can the set $\{a, b, c, d, e, f, g\}$ be partitioned into two three-element sets and one single-element set such that $\{a\}$ is missing?
(1) $45$ (2) $50$ (3) $56$ (4) $60$
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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) $220$
%% Page 9
Download of Descriptive Exam Questions and Answers from Riazisara Website
ریاضیات 121-A صفحه ۸

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

150- At minimum, how many subsets must be chosen from the set $\{7, \ldots, 3, 2, 1\}$ so that we are certain that two subsets share a common element?
(4) $46$ (3) $45$ (2) $64$ (1) $65$
Place for Calculations
%% Page 10
Control Code: 122 A
Download of questions and descriptive answer keys of the national entrance exam from the Riazi Sara website
www.riazisara.ir
National University Entrance Exam for Universities and Higher Education Institutions of the Country --- Year 1401
Mathematical and Technical Sciences Group Specialized Exam

Notes	Response Time	To Question No.	From Question No.	Number of Questions	Subject
70 questions	50 minutes	190	151	40	Physics
80 minutes	30 minutes	220	191	30	Chemistry

%% Page 11 Physics $\leftarrow$ 122-A $\rightarrow$ Page 2

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

Let $A_{m,n}$ denote the set of strictly increasing sequences $1 \leq \alpha_1 < \alpha_2 < \cdots < \alpha_m \leq n$ of integers, $B_{m,n}$ denote the set of non-negative integer solutions of $\alpha_1 + \alpha_2 + \cdots + \alpha_m = n$, and $C_{m,n}$ denote the set of strictly increasing sequences chosen from $\{1,2,\ldots,n\}$.
a) Construct a bijection from $A_{m,n}$ to $B_{m+1,n-1}$.
b) Construct a bijection from $A_{m,n}$ to $C_{m,m+n-1}$.
c) Find the number of elements in $A_{m,n}$.

In how many ways can the numbers $1, 2, \ldots, 9$ be arranged in a $3 \times 3$ grid \[ \begin{array}{|c|c|c|} \hline A & B & C \\ \hline D & E & F \\ \hline G & H & I \\ \hline \end{array} \] such that each row and each column is in increasing order (i.e., $A < B < C$, $D < E < F$, $G < H < I$, $A < D < G$, $B < E < H$, $C < F < I$)?

There are 8 balls numbered $1,2 , \ldots , 8$ and 8 boxes numbered $1,2 , \ldots , 8$. The number of ways one can put these balls in the boxes so that each box gets one ball and exactly 4 balls go in their corresponding numbered boxes is
(a) $3 \times {}^{8}\mathrm{C}_{4}$
(b) $6 \times {}^{8}\mathrm{C}_{4}$
(c) $9 \times {}^{8}\mathrm{C}_{4}$
(d) $12 \times {}^{8}C_{4}$

Let $A$ be the set $\{ 1,2 , \ldots , 20 \}$. Fix two disjoint subsets $S _ { 1 }$ and $S _ { 2 }$ of $A$, each with exactly three elements. How many 3-element subsets of $A$ are there, which have exactly one element common with $S _ { 1 }$ and at least one element common with $S _ { 2 }$?
(a) 51
(b) 102
(c) 135
(d) 153

In how many ways can $10$ A's and $6$ B's be arranged in a row such that no two B's are adjacent and no two A's are adjacent? (More precisely: find the number of arrangements of $10$ A's and $6$ B's such that between any two consecutive B's there is at least one A, and between any two consecutive A's there is at least one B.)

Let $S = \{ 1, 2, \ldots, n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2^n$
(B) $3^n$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n!$

The number of maps $f$ from the set $\{ 1, 2, 3 \}$ into the set $\{ 1, 2, 3, 4, 5 \}$ such that $f(i) \leq f(j)$ whenever $i < j$ is
(A) 60
(B) 50
(C) 35
(D) 30

Consider three boxes, each containing 10 balls labelled $1, 2, \ldots, 10$. Suppose one ball is drawn from each of the boxes. Denote by $n_i$, the label of the ball drawn from the $i$-th box, $i = 1, 2, 3$. Then the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is
(A) 120
(B) 130
(C) 150
(D) 160

A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(A) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$

The number of ways in which one can select six distinct integers from the set $\{1, 2, 3, \cdots, 49\}$, such that no two consecutive integers are selected, is
(A) $\binom{49}{6} - 5\binom{48}{5}$
(B) $\binom{43}{6}$
(C) $\binom{25}{6}$
(D) $\binom{44}{6}$

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

Let $S = \{ 1,2 , \ldots , n \}$. Find the number of unordered pairs $\{ A , B \}$ of subsets of $S$ such that $A$ and $B$ are disjoint, where $A$ or $B$ or both may be empty.

Let $S = \{ 1,2 , \ldots , n \}$. Find the number of unordered pairs $\{ A , B \}$ of subsets of $S$ such that $A$ and $B$ are disjoint, where $A$ or $B$ or both may be empty.

The number of maps $f$ from the set $\{ 1,2,3 \}$ into the set $\{ 1,2,3,4,5 \}$ such that $f ( i ) \leq f ( j )$ whenever $i < j$ is
(a) 60
(b) 50
(c) 35
(d) 30

The number of maps $f$ from the set $\{ 1,2,3 \}$ into the set $\{ 1,2,3,4,5 \}$ such that $f ( i ) \leq f ( j )$ whenever $i < j$ is
(a) 60
(b) 50
(c) 35
(d) 30

Let $S = \{ 1, 2, \ldots, n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2 ^ { n }$
(B) $3 ^ { n }$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n !$