This is beta site. https://beta.mathcity.org/ 2026-06-09T20:46:08+00:00 https://beta.mathcity.org/ https://beta.mathcity.org/_media/logo.png text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-4-p4?rev=1737476038&do=diff Question 4 Exercise 6.4 Solutions of Question 4 of Exercise 6.4 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.\begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\\ \text{then} n(S)&=2^3=8\end{align}$$A=\{H H H\}$$$$n(A)=1$$$P(A)=\dfrac{n(A)}{n(S)}=\dfrac{1}{8}$\begin{align}S&=(HHII,HHT.HTH.HTT.THII.THT.TTH,TT7),\\ \text{then} n(S)&=2^3=8\end{align}… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-1-p2?rev=1737476038&do=diff Question 3 & 4 Exercise 6.1 Solutions of Question 3 & 4 of Exercise 6.1 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{1}{6 !}+\dfrac{2}{7 !}+\dfrac{3}{8 !}=\dfrac{75}{8 !}$\begin{align}\dfrac{1}{6 !}+\dfrac{2}{7 !}+\dfrac{3}{8 !}&=\dfrac{1}{6 !}+\dfrac{2}{7.6 !}+\dfrac{3}{8.7 .6 !} \\ & =\dfrac{56+16+3}{8 !}\\ &=\dfrac{75}{8 !}\end{align}$\df… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.5 Solutions of Question 1 and 2 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$A$$B$$P(A)=\dfrac{2}{5}, P(B)=\dfrac{2}{5}$$P(A \cup B)=\dfrac{1}{2}$$P(A \cap B)$\begin{align} P(A \cup B)&=P(A)+P(B)-P(A \cap B) \\ \Rightarrow P(A \cap B)&=P(A)+P(B)-P(A \cup B) \end{align}$P(A), P(B)$$P(A \cup B)$$$P(A \cap… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p1?rev=1737476038&do=diff Question 1 and 2 Exercise 6.2 Solutions of Question 1 and 2 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^6 P_6$\begin{align}^6 P_6&=\dfrac{6 !}{(6-6) !}\\ &=6 !=720\end{align}$^{20} P_2$\begin{align}^{20} P_2&=\dfrac{20 !}{(20-2) !}\\ &=\dfrac{20.19 .18 !}{18 !}\\ &=20 \times 19=380\end{align}$^{16} P_3$\begin{align}^{16} P_3&=\dfr… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p1?rev=1737476038&do=diff Question 1 Exercise 6.3 Solutions of Question 1 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^n C_2=36$$n$\begin{align}&^n C_2=36\\ & \Rightarrow \dfrac{n !}{(n-2) ! 2 !}=36 \\ & \Rightarrow \dfrac{n(n-1)(n-2) !}{(n-2) ! \cdot 2}=36 \\ & \Rightarrow n(n-1)=72 \\ & \Rightarrow n^2-n-72=0 \\ & \Rightarrow n^2-9 n+8 n-72=0\\ & \Rightar… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 6.2 Solutions of Question 3 and 4 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$^n P_r=n(^{n-1} P_{r-1})$$$^n P_r=n({ }^{n-1} P_{r-1})$$\begin{align}n(^{n-1} P_{r-1})&=n \dfrac{(n-1) !}{((n-1)-(r-1)) !} \\ & =\dfrac{n(n-1) !}{(n-r) !}\\ &=\dfrac{n !}{(n-r) !}\\ &=^n P_r\end{align}$^n P_r=^{n-1} P_r+r(^{n-1} … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p4?rev=1737476038&do=diff Question 4 Exercise 6.3 Solutions of Question 4 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{n-1} C_r+{ }^{n-1} C_{r-1}={ }^n C_r$$${ }^n{ }^1 C_r+{ }^n{ }^1 C_{r-1}={ }^n C_s$$\begin{align} { }^{n-1} C_r+{ }^{n-1} C_{r-1}&=\dfrac{(n-1) !}{(n-r-1) ! r !}+\dfrac{(n-1) !}{(n-1-(r-1)) !(r-1) !} \\ & =\dfrac{(n-1) !}{(n-r-1) ! r(r-… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p2?rev=1737476038&do=diff Question 3 and 4 Exercise 6.5 Solutions of Question 3 and 4 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$P(A)=0.5$$P(A \cup B)=0.6$$P(B)$$A$$B$$\mathrm{A}$$B$$A \cap B=\emptyset$\begin{align}P(A \cup B)&=P(A)+P(B)\\ \Rightarrow P(B)&=P(A \cup B)-P(A)\\ &=0.6-.0 .5=0.1 \end{align}$30$$1$$30.$\begin{align}S&=\{1,2,3, \ldots, 50\} \tex… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-5-p3?rev=1737476038&do=diff Question 5 and 6 Exercise 6.5 Solutions of Question 5 and 6 of Exercise 6.5 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$\dfrac{8}{9}$$$E=\{ event\, passing\, the\, test \}$$$$E^{\prime}=\{ event\, failing\, the\, test \}$$$E$$E^{\prime}$$P(E)=\dfrac{8}{9}$\begin{align}P(E^{\prime})&=1-P(E)=1-\dfrac{8}{9}=\dfrac{1}{9}\end{align}$4$$4$\begin{align}S… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p2?rev=1737476038&do=diff Question 2 Review Exercise 6 Solutions of Question 2 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{2 n} C_r={ }^{2 n} C_{r+2}$$r$\begin{align} { }^{2 n} C_r&={ }^{2 n} C_{r+2} \\ \Rightarrow \dfrac{(2 n) !}{(2 n-r) ! r !}&=\dfrac{(2 n) !}{(2 n-(r+2)) !(r+2) !}\end{align}$(2 n)$\begin{align} \Rightarrow \dfrac{1}{(2 n-r) ! r… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-2-p5?rev=1737476038&do=diff Question 9 Exercise 6.2 Solutions of Question 9 of Exercise 6.2 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$=^6 P_1=6$$s=^6 P_2=30$$=^6 P_3=120$$=^6 P_4=360$$=^6 P_5=720$$=^6 P_6=720$$6+30+120+360+720+720=1956$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p2?rev=1737476038&do=diff Question 2 Exercise 6.3 Solutions of Question 2 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$n$$r$${ }^n P_r=840$${ }^n C_r=35$\begin{align} &^n P_r=\dfrac{n !}{(n-r) !}=840 ....(i)\\ &^n C_r=\dfrac{n !}{(n-r) ! r !}=35....(ii)\end{align}\begin{align}\dfrac{n !}{(n-r) !} \cdot \dfrac{(n-r) ! r !}{n !}&=\dfrac{840}{35}\\ r!&=24\\ \te… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/ex6-3-p5?rev=1737476038&do=diff Question 5 and 6 Exercise 6.3 Solutions of Question 5 and 6 of Exercise 6.3 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$12$$n=12$${ }^{12} C_2=66$$12$$n=12$${ }^{12} C_3=220$$${ }^6 C_2=\dfrac{6 !}{(6-2) ! 2 !}=15 $$$6$$\quad 15-6=9$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p3?rev=1737476038&do=diff Question 3 & 4 Review Exercise 6 Solutions of Question 3 & 4 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.${ }^{56} P_{r+6}:{ }^{54} P_{r+3}=30800: 1$$r$\begin{align} { }^{56} P_{r+6}:{ }^{54} P_r+3&=30800: 1 \\ \Rightarrow \dfrac{\dfrac{56 !}{[56-(r+6)] !}}{\dfrac{54 !}{[54-(r+3)] !}}&=\dfrac{30800}{1} \\ \Rightarrow \dfrac{56… text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p5?rev=1737476038&do=diff Question 7 & 8 Review Exercise 6 Solutions of Question 7 & 8 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$P(A)=0.8, P(B)=0.5$$P(B / A)=0.4$$P(A \cap B)$\begin{align} P(B \mid A)&=\dfrac{P(A \cap B)}{P(A)} \\ \Rightarrow P(A \cap B)&=P(B \mid A) \cdot P(A)\\ &=0.4 \times 0.8=0.32\end{align}$P(A)=0.8, P(B)=0.5$$P(B / A)=0.4$$P(A … text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p6?rev=1737476038&do=diff Question 9 & 10 Review Exercise 6 Solutions of Question 9 & 10 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$2,3,0,3,4,2,3$$1$$=100,0000$$$=\dfrac{7 !}{3 ! \cdot 2 !}=420 $$$1$$0$$7$$0$$$=\dfrac{6 !}{2 ! 3 !}=60 $$$1$$420-50=360$$n$$n$$(n-1)$$(n - 1)$$(n-1)$$(n-2) !$$2$$2 !$$n$$$(n-2) ! \cdot 2 !=2(n-2) ! $$ text/html 2025-01-21T16:13:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/math-11-kpk/sol/unit06/re-ex6-p7?rev=1737476038&do=diff Question 11 Review Exercise 6 Solutions of Question 11 of Review Exercise 6 of Unit 06: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.$$n(S)=4$$$\dfrac{1}{4}$$$\quad P( orange )=\dfrac{1}{4}$$$\dfrac{1}{4}$$\dfrac{1}{4}$\begin{align}P(\operatorname{Red})&=\dfrac{1}{4}\\ P( Green )&=\dfrac{1}{4}\end{align}$P(R \cap G)=\phi$$R$$G$\begin{align}\boldsymbol{P}( Red o…