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- Question 13 Exercise 6.2
- word "Excellence." How many of these permutations begin with $\mathrm{E}$ ? ====Solution==== The total... m_2=2$ are $L$ and $m_3=2$ are $C$. Therefore, \begin{align}\text{total number of permutations are} &=\left(\begin{array}{c} n \\ m_1, m_2, m_3 \end{array}\right)\\&=\left(\begin{array}{c} 10 \\ 4,2,2 \end{array}\right) \\ &
- Question 9 Exercise 6.3
- Question 9(i)===== An $8$-persons committee is to be formed from a group of $6$ women and $7$ men. In how many ways can the committee be chosen if (i) committee must contain four men and... 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 are ${ }^6 C_4$. By fundamental princip
- Question 7 Exercise 6.4
- n==== The sample space rolling a pair of dice is \begin{align}S&=\{(i, j) ; i, j=1,2,3,4,5,6\}\\ &=\left[\begin{array}{llllll} (1,1) & (1,2) & (1,3) & (1,4) &... )=6 \times 6=36$$ doublet of even numbers. Let \begin{align}A&=\{(2,2),(4,4),(6,6)\}\\ n(A)&=3\end{a... n==== The sample space rolling a pair of dice is \begin{align} S&=\{(i, j) ; i, j=1,2,3,4,5,6\}\\ &=\l
- Question 9 Exercise 6.5
- $2$ vacancies. The probability of their selection being $\dfrac{1}{7}$ and $\dfrac{1}{5}$ respectively. Find the probability that both will be selected. ====Solution==== \begin{align} P(\text { Ajmal scicction })&=\dfrac{1}{7} \\ \Rightarrow P(... therefore these two events are independent. Thus \begin{align}P(\text { Both are selected })&=P(A) \ti
- Question 12 Exercise 6.2
- ==Question 12(i)===== How many different word can be formed from the letters "BOOKWORM" if the letters... mber of different words using all at a time are: \begin{align} \left(\begin{array}{c} n \\ m 1 \end{array}\right)&=\left(\begin{array}{l} 8 \\ 3 \end{array}\right) \\ & =\dfr
- Question 7 and 8 Exercise 6.2
- stion 7(i)===== How many three digits numbers can be formed from the digits $1,2,3,4$ and 5 if repetit... hundred digit, ten digit and unit digit place) to be filled by five digits, Moreover repetition is a... tion 7(ii)===== How many three digits numbers can be formed from the digits $1,2,3,4$ and 5 if repetit... estion 8===== How many different arrangements can be formed of the word "equation" if all the vowels a
- Question 10 Exercise 6.2
- ion 10(i)===== In how many ways can five students be seated in a row of eight seals if a certain two s... The total number of ways these five students can be seated are: \begin{align}^8 P_5&=\dfrac{8 !}{(8-5) !}\\ &=\dfrac{8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \c... t next to each other then these two students will be handled as a single students and the eights seats
- Question 1 Exercise 6.4
- =====Question 1(a)===== Let $S=\{1,2,3,4,5,6\}$ be the sample space of rolling a dice. What is the p... lling a $5$ ? ====Solution==== Rolling a $5$ Let \begin{align}A&=\{5\}\\ P(A)&=\dfrac{n(A)}{n(S)}\\ &=... =====Question 1(b)===== Let $S=\{1,2,3,4,5,6\}$ be the sample space of rolling a dice. What is the p... =Solution==== Rolling a number less than $1$ Let \begin{align}B&=\{\}\\ &=\phi \text{then}\\ P(B)&=\df
- Question 1 and 2 Exercise 6.2
- ion 1(i)===== Evaluate $^6 P_6$ ====Solution==== \begin{align}^6 P_6&=\dfrac{6 !}{(6-6) !}\\ &=6 !=720... 1(ii)===== Evaluate $^{20} P_2$ ====Solution==== \begin{align}^{20} P_2&=\dfrac{20 !}{(20-2) !}\\ &=\d... (iii)===== Evaluate $^{16} P_3$ ====Solution==== \begin{align}^{16} P_3&=\dfrac{16 !}{(16-3) ! }\\ &=\... ^n P_3)$ for $n.$ ====Solution==== We are given: \begin{align}^n P_5&=56(^n P_3) \\ \Rightarrow \dfrac
- Question 2 Exercise 6.3
- d ${ }^n C_r=35$. ====Solution==== We are given: \begin{align} &^n P_r=\dfrac{n !}{(n-r) !}=840 ....(i... 5....(ii)\end{align} Dividing Eq.(i) by Eq.(ii) \begin{align}\dfrac{n !}{(n-r) !} \cdot \dfrac{(n-r) ... &=4\end{align} Putting $r=4$ in Eq.(ii), we get \begin{align} & { }^n C_4=\dfrac{n !}{(n-4) ! 4 !}=35... ign} Let $y=n^2-3 n$ then the above last equation becomes \begin{align} & y(y+2)=840 \\ & \Rightarrow
- Question 4 Exercise 6.4
- Hence the sample space of the given problem is: \begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\... Hence the sample space of the given problem is: \begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\... Hence the sample space of the given problem is: \begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\... Hence the sample space of the given problem is: \begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\
- Question 1 and 2 Exercise 6.1
- frac{10 !}{3 ! .3 ! \cdot 4 !}$ ====Solution==== \begin{align}\dfrac{10 !}{3 ! \cdot 3 ! \cdot 4 !}&=\... the $\dfrac{3 !+4 !}{5 !-4 !}$ ====Solution==== \begin{align}\dfrac{3 !+4 !}{5 !-4 !}&=\dfrac{3 !+4.3... the $\dfrac{(n-1) !}{(n+1) !}$ ====Solution==== \begin{align}\dfrac{(n-1) !}{(n+1) !} &= \dfrac{(n-1... ate the $\dfrac{10 !}{(5 !)^2}$ ====Solution==== \begin{align}\dfrac{10 !}{(5 !)^2}&=\dfrac{10 \cdot 9
- Question 1 and 2 Exercise 6.5
- ution==== We know by addition law of probability \begin{align} P(A \cup B)&=P(A)+P(B)-P(A \cap B) \\ ... mplementary events $$P(B)=1-P(\bar{B})$$ Putting \begin{align}P(\bar{B})&=\dfrac{5}{8}\\ P(B)&=1-\dfra... })=1-P(A)$$ Putting $P(A)=\dfrac{1}{2}$, we get \begin{align}P(\bar{A})&=1-\dfrac{1}{2}=\dfrac{1}{2}\... w by addition law of probability, we know that: \begin{align}P(A \cap B)&=P(A)+P(B)-P(A \cap B) \\ \R
- Question 5 and 6 Exercise 6.2
- In how many ways can letter of the word 'Fasting' be arranged? ====Solution==== The total number of al... nts to fill $7$ places by these $7$ letters are: \begin{align}^7 P_7&=\dfrac{7 !}{(7-7) !}\\ & =7 !\\ ... ==Question 6===== How many four digits number can be formed from the digits $2,4,5,7,9$ ? (Repetition not being allowed). How many of these are even? ====Solu
- Question 1 Review Exercise 6
- ollapse> ii. How many two digits odd numbers can be formed form the digits $\{1,2,3,4,5,6,7\}$ if rep... </collapse> iii. How many six digits number can be formed from the digits $\{1,2,3,4,6,7,8\}$ withou... e> v. In how many different ways can $5$ couples be seated around a circular table if the couple must not be separated? * (a) $768$ * %%(b)%% $724