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- Definitions: FSc Part 1 (Mathematics): PTB @fsc-part1-ptb
- ptb:definitions-aurang-zaib]] ===== Chapter 01: Number system ===== * **Rational number:** A number which can be written in the form of $\frac{p}{q}$, where $p,q \in \mathbb{Z}$, $q\neq 0$, is called a rational number * **Irrational number:** A real number which c
- Question 9 Exercise 6.3 @math-11-kpk:sol:unit06
- re $7$ and total women are $6.$ Therefore, Total number of persons $=7+6=13$ Committee consist of 8 pers... e contain exactly four men and four women. Total number of different ways that four men to be selected are: ${ }^7 C_4$. Total number of different ways that four women to be selected ... . By fundamental principle of counting the total number of different committees that will exactly contain
- Question 7 Exercise 6.4 @math-11-kpk:sol:unit06
- 5) & (6,6) \end{array}\right]\end{align} So total number of sample points are $$n(S)=6 \times 6=36$$ doub... nce the possibility of getting doublet of an even number is: $$P(A)=\dfrac{n(A)}{n(S)}=\dfrac{3}{36}=\dfra... 5) & (6,6) \end{array}\right]\end{align} So total number of sample points are $$n(S)=6 \times 6=36$$ A sum less than $6$ Let $B=\{$ a number less than 6$\}$, then from sample space, we see
- Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib @fsc-part1-ptb
- for his valuable contribution. =====Chapter 01: Number System===== ====Rational Number==== A number which can be expressed in the form \( \dfrac{p}{q} \), where \( p, q \in \mathbb{Z} \) and \( q \neq 0 \), is termed as a rational number. ===Example==== \( \dfrac{3}{4} \), \( \dfrac{7}{
- Question 13 Exercise 6.2 @math-11-kpk:sol:unit06
- war, Pakistan. =====Question 13(i)===== Find the number of permutation of word "Excellence." How many of ... in with $\mathrm{E}$ ? ====Solution==== The total number of letters in 'Excellence' are: $n=10$, out of wh... 2$ are $C$. Therefore, \begin{align}\text{total number of permutations are} &=\left(\begin{array}{c} n ... d $m_3=2$ are $C$. Therefore, \begin{align}\text{Number of permulations are} &=\left(\begin{array}{c} n \
- Algebraic Number Theory Notes by Anwar Khan @notes
- ====== Algebraic Number Theory Notes by Anwar Khan ====== {{ :notes:algebraic-number-theory-notes-anwar-khan.jpg?nolink|Algebraic Number Theory Notes by Anwar Khan}} Algebraic number theory is a subfield of number theory that studies integers
- Number Theory: Handwritten Notes @notes
- ====== Number Theory: Handwritten Notes ====== {{ :notes:number-theory-handwritten-notes.jpg?nolink&600|Number Theory: Handwritten Notes}} The study of the characteristi... of the positive integers (1, 2, 3,...) is called number theory. It is significant because it has numerous
- Number Theory by Dr Muhammad Umer Shuaib @notes
- ====== Number Theory by Dr Muhammad Umer Shuaib ====== {{ :notes:number-theory-muzammil-tanveer.jpg?nolink&600|Number Theory Notes}} A subfield of mathematics called number theory studies the characteristics of positive int
- MTH321: Real Analysis I (Spring 2023) @atiq
- om Chapter 02** - A convergent sequence of real number has one and only one limit (i.e. limit of the seq... }$ also converges to $s$. - For each irrational number $x$, there exists a sequence $\left\{ {{r}_{n}} \... _{n}}}$ is convergent if and only if for any real number $\varepsilon >0$, there exists a positive integer... r all $n$. Also suppose that for a fixed positive number $\lambda $ and positive integer $k$, $a_n<\lambda
- Number Theory Notes by Anwar Khan @notes
- ====== Number Theory Notes by Anwar Khan ====== {{ :notes:number-theory-notes-anwar-khan.jpg?nolink|Number Theory Notes by Anwar Khan}} Mathematicians who specialize in number theory examine the characteristics and connection
- Notes of Mathematics
- dwritten notes by Mr Raheel Ahmad. </well> ===== Number Theory ===== <well> **[[Notes:algebraic-number-theory-notes-anwar-khan]]** \\ A handwritten notes of the Algebraic Number Theory by Mr. Anwar Khan. These notes covers many topics of the subject. </well> <well> **[[Notes:number-theory-handwritten-notes]]** \\ A small and compr
- Question 7 and 8 Exercise 6.2 @math-11-kpk:sol:unit06
- us by fundamental principle of counting the total number of three digits in this case are: $$m_1 \cdot m_2... t allowed then each digit can appear once in each number. In this case $E_1$ occurs in $m_1=5$ different... s by fundamental principle of 'counting the total number of three digits in this case are: $$m_1 \cdot m_2... e to be kept together? ====Solution==== The total number of alphabets in word equation are $8$, out of whi
- Question 9 Exercise 6.2 @math-11-kpk:sol:unit06
- e given by six flags of different colors when any number of them used at a time? ====Solution==== We have ... . If each signal consist of one color then total number of signals $=^6 P_1=6$. If each signal consist of two color then total number of signal $s=^6 P_2=30$. If each signal consist of three color then total number of signals $=^6 P_3=120$. If each signal consist
- Question 10 Exercise 6.2 @math-11-kpk:sol:unit06
- itting next to each other? ====Solution==== Total number of seats are eight, so $n=8$. Number of students are five so, $r=5$. The total number of ways these five students can be seated are: \begin{a... will be considered as 7. In this case the total number of ways are: \begin{align}^2 P_2 \times^7 P_4&=2
- Question 12 Exercise 6.2 @math-11-kpk:sol:unit06
- at a time? ====Solution==== BOOKWORM\\ The total number of letters in word BOOKWORM are $8.$ $n=8$ out o... hree are $\mathrm{O}$, so $m_1=3$.. Thus total number of different words using all at a time are: \begi... t a time? ====Solution==== BOOKKEEPER\\ The total number of letters in $\mathrm{BOOK}$ KEEPER are ten. $n... wo are $\mathrm{K}$, so $m_3=2$. Thus the total number of different words are: \begin{align} \left(\begi