LFM Stats And Pure

iran-konkur 2014 Q144 Subset Counting with Set-Theoretic Conditions View

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) $32$

iran-konkur 2014 Q145 Basic Combination Computation View

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) $20$

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iran-konkur 2016 Q146 Basic Combination Computation View

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) $15$

iran-konkur 2018 Q146 Partitioning into Teams or Groups View

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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iran-konkur 2018 Q153 Counting Integer Solutions to Equations View

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

iran-konkur 2019 Q147 Counting Integer Solutions to Equations View

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

iran-konkur 2020 Q143 Counting Integer Solutions to Equations View

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

iran-konkur 2022 Q150 Pigeonhole Principle Application View

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

isi-entrance None Q1 Counting Integer Solutions to Equations View

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

isi-entrance 2005 Q7 Combinatorial Identity or Bijection Proof View

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 $A_{m,n}$.

isi-entrance 2007 Q8 Permutation Properties and Enumeration (Abstract) View

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., $A < B < C$, $D < E < F$, $G < H < I$, $A < D < G$, $B < E < H$, $C < F < I$)?

isi-entrance 2010 Q1 Distribution of Objects into Bins/Groups View

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) $12 \times {}^{8}C_{4}$

isi-entrance 2011 Q6 Selection with Group/Category Constraints View

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

isi-entrance 2012 Q21 Linear Arrangement with Constraints View

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.)

isi-entrance 2013 Q25 4 marks Subset Counting with Set-Theoretic Conditions View

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) $n!$

isi-entrance 2013 Q26 4 marks Counting Functions or Mappings with Constraints View

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

isi-entrance 2013 Q27 4 marks Counting Functions or Mappings with Constraints View

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

isi-entrance 2013 Q40 4 marks Selection with Group/Category Constraints View

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) $\binom { 10 } { 3 } \binom { 14 } { 4 }$

isi-entrance 2013 Q59 4 marks Selection with Adjacency or Spacing Constraints View

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) $\binom{44}{6}$

isi-entrance 2014 Q10 Counting Integer Solutions to Equations View

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

isi-entrance 2015 QB7 Subset Counting with Set-Theoretic Conditions View

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.

isi-entrance 2015 QB7 Subset Counting with Set-Theoretic Conditions View

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.

isi-entrance 2015 Q16 4 marks Counting Functions or Mappings with Constraints View

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

isi-entrance 2015 Q16 4 marks Counting Functions or Mappings with Constraints View

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

isi-entrance 2016 Q25 4 marks Subset Counting with Set-Theoretic Conditions View

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) $n !$

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LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

Solving quadratics and applications 106

Curve Sketching 295

Factor & Remainder Theorem 90

Polynomial Division & Manipulation 44

Binomial Theorem (positive integer n) 244

Function Transformations 65

Composite & Inverse Functions 249

Partial Fractions 22

Generalised Binomial Theorem 16

Generalised Binomial Theorem and Partial Fractions 3

Complex Numbers Arithmetic 283

Complex Numbers Argand & Loci 135

Permutations & Arrangements 276

Combinations & Selection 268

Measures of Location and Spread 233

Probability Definitions 413

Tree Diagrams 5

Principle of Inclusion/Exclusion 23

Independent Events 51

Conditional Probability 127

Discrete Probability Distributions 210

Binomial Distribution 107

Geometric Distribution 11

Hypergeometric Distribution 10

Negative Binomial Distribution 2

Modelling and Hypothesis Testing 38

Hypothesis test of binomial distributions 12

Data representation 29

Continuous Probability Distributions and Random Variables 30

Continuous Uniform Random Variables 15

Geometric Probability 28

Normal Distribution 96

Approximating Binomial to Normal Distribution 7

Sign Change & Interval Methods 74

Fixed Point Iteration 12