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- Exercise 6.2 (Solutions)
- given on this page. This exercise consists of the question related to factorial function. The book misses t... t{ if } n=0. \end{array} \right. $$ </callout> **Question 1.** Prove the following for $n \in \mathbb{N}$.\... -2}$\\ [[math-11-nbf:sol:unit06:ex6-2-p1|Solution Question 1]] **Question 2.** Find $n$, if:\\ (i) $\quad n P_4=20^n P_2$ (ii) ${ }^{2 n} P_3=100{ }^n P_2$ (iii) $16
- Exercise 6.3 (Solutions)
- given on this page. This exercise consists of the question related to factorial function. **Question 1(i-v).** Prove the following for $n \in \mathbb{N}$.\\ (i) ${ ... r}$\\ [[math-11-nbf:sol:unit06:ex6-3-p1|Solution: Question 1(i-v)]] **Question 1(vi-x).** Prove the following for $n \in \mathbb{N}$.\\ (vi) ${ }^{2 n} \mathrm{C}_{\
- Exercise 6.1 (Solutions)
- given on this page. This exercise consists of the question related to factorial function. The book misses t... t{ if } n=0. \end{array} \right. $$ </callout> **Question 1.** Evaluate the following:\\ (i) $10!$ (ii) $\d... $ \\ [[math-11-nbf:sol:unit06:ex6-1-p1|Solution: Question 1]] **Question 2.** Write the following in factorial form:\\ (i) 14.13 .12 .11 (ii) 1.3.5.7.9 (iii) $n\lef
- Question 2, Exercise 6.2
- ====== Question 2, Exercise 6.2 ====== Solutions of Question 2 of Exercise 6.2 of Unit 06: Permutation and Combination. Thi... deral Textbook Board, Islamabad, Pakistan. =====Question 2(i)===== Find $n$, if: $\quad ^nP_4=20\, ^nP_2$ ... n$ should be a positive integer, so $n=7$\\ =====Question 2(ii)===== Find $n$, if: $\quad ^{2n}P_3=100 \, ^
- Review Exercise (Solutions)
- Islamabad, Pakistan are given on this page. **Question 1.** Select the best matching option. Choose the ... . \\ [[math-11-nbf:sol:unit06:Re-ex6-p1|See MCQs: Question 1]] **Question 2.** How many words can be formed by using $4$ distinct alphabets?\\ [[math-11-nbf:sol:unit06:Re-ex6-p2|Solution: Question 2 & 3 ]] **Question 3.** How many $3$-digit numb
- Question 3, Exercise 6.2
- ====== Question 3, Exercise 6.2 ====== Solutions of Question 3 of Exercise 6.2 of Unit 06: Permutation and Combination. Thi... deral Textbook Board, Islamabad, Pakistan. =====Question 3(i)===== Find $r$, if: $^6P_{r-1}=^5P_4$ ** Sol... 7-r)!\\ 3&=7-r \\ r&=7-3\\ r&=4\end{align*} =====Question 3(ii)===== Find $r$, if: $^{10}P_{r}=2\,^9P_r$ *
- Question 7(i-vi), Exercise 6.1
- ====== Question 7(i-vi), Exercise 6.1 ====== Solutions of Question 7(i-vi) of Exercise 6.1 of Unit 06: Permutation and Comb... deral Textbook Board, Islamabad, Pakistan. =====Question 7(i)===== Find $n$, if $\quad \dfrac{n!}{(n-2)!}=... \end{align*} Since $$n\geq 2\implies n=31$$ =====Question 7(ii)===== Find $n$, if $\quad \dfrac{n!}{(n-5)!}
- Question 2, Exercise 6.3
- ====== Question 2, Exercise 6.3 ====== Solutions of Question 2 of Exercise 6.3 of Unit 06: Permutation and Combination. Thi... deral Textbook Board, Islamabad, Pakistan. =====Question 2(i)===== Find $n$, if : $\,\, ^nC_5=\,\, ^nC_8$ ... ning roots of $n$ are imaginary so $n=3$.\\ =====Question 2(ii)===== Find $n$, if : $\,\, ^nC_{15}=\,\, ^nC
- Question 6(i-v), Exercise 6.1
- ====== Question 6(i-v), Exercise 6.1 ====== Solutions of Question 6(i-v) of Exercise 6.1 of Unit 06: Permutation and Combin... deral Textbook Board, Islamabad, Pakistan. =====Question 6(i)===== Prove for $n\in N$: $\quad (2n)!=2^n(n!... 1.3.5.\cdots(2n-1))\\ &=R.H.S. \end{align*} =====Question 6(ii)===== Prove for $n\in N$: $\quad (n+1)[n!n+(
- Question 1(vi-x), Exercise 6.3
- ====== Question 1(vi-x), Exercise 6.3 ====== Solutions of Question 1(vi-x) of Exercise 6.3 of Unit 06: Permutation and Comb... deral Textbook Board, Islamabad, Pakistan. =====Question 1(vi)===== Prove for $n\in N$: $\quad^{2n}C_n=\df... ot 5 \ldots(2 n-1))}{n!}= R.H.S\end{align*} =====Question 1(vii)===== Prove for $n\in N$: $\quad^nC_p=^nC_
- Question 7 and 8, Exercise 6.3
- ====== Question 7 and 8, Exercise 6.3 ====== Solutions of Question 7 and 8 of Exercise 6.3 of Unit 06: Permutation and Comb... deral Textbook Board, Islamabad, Pakistan. =====Question 7(i)===== A committee of $5$ members is to be for... _{2} \times{ }^{6} C_{3}=6 \times 20=120$\\ =====Question 7(ii)===== A committee of $5$ members is to be fo
- Question 13 and 14, Exercise 6.3
- ====== Question 13 and 14, Exercise 6.3 ====== Solutions of Question 13 and 14 of Exercise 6.3 of Unit 06: Permutation and ... deral Textbook Board, Islamabad, Pakistan. =====Question 13===== In a examination, a candidate has to pass... \\ $$\text{Total}\quad =6+15+20+15+6+1=63$$ =====Question 14===== A question papers has three parts $A$, $B
- Question 1, Exercise 6.1
- ====== Question 1, Exercise 6.1 ====== Solutions of Question 1 of Exercise 6.1 of Unit 06: Permutation and Combination. Thi... deral Textbook Board, Islamabad, Pakistan. =====Question 1(i)===== Evaluate $10!$. ** Solution. ** \begi... times 5 \times 4 \\ &= 3628800 \end{align*} =====Question 1(ii)===== Evaluate $\dfrac{12!}{7! 3! 2!}$. **
- Question 2, Exercise 6.1
- ====== Question 2, Exercise 6.1 ====== Solutions of Question 2 of Exercise 6.1 of Unit 06: Permutation and Combination. Thi... deral Textbook Board, Islamabad, Pakistan. =====Question 2(i)===== Write in the fractional form: $\quad 14... {10!} \\ = & \dfrac{14!}{10!} \end{align*} =====Question 2(ii)===== Write in the fractional form: $\quad 1
- Question 3 and 4, Exercise 6.1
- ====== Question 3 and 4, Exercise 6.1 ====== Solutions of Question 3 and 4 of Exercise 6.1 of Unit 06: Permutation and Comb... deral Textbook Board, Islamabad, Pakistan. =====Question 3(i)===== Prove that: $\quad \dfrac{1}{5!}+\dfrac... & \dfrac{4}{315} = RHS \end{align*} GOOD =====Question 3(ii)===== Prove that: $\quad \dfrac{(n-1)!}{(n-3