Exercise 6.3 (Solutions)

The solutions of the Exercise 6.3 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are 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) ${ }^{n} C_{r}=\frac{n!}{r!(n-r)!}$ (ii) $n,{ }^{n-1} C_{r-1}=(n-r+1){ }^{n} C_{r-1}$ (iii) $r^{n} C_{r}=(n-r+1)^{n} C_{r-1}$
(iv) ${ }^{n-1} C_{r-1}+{ }^{n-1} C_{r}={ }^{n} C_{r}$ (v) ${ }^{n} C_{r}+{ }^{n} C_{r-1}={ }^{n+1} C_{r}$
Solution: Question 1(i-v)

Question 1(vi-x). Prove the following for $n \in \mathbb{N}$.
(vi) ${ }^{2 n} \mathrm{C}_{\mathrm{n}}=\frac{2^{n} \cdot[1.3 .5 \ldots(2 n-1)]}{n!}$ (vii) ${ }^{n} C_{p}={ }^{n} C_{q} \Rightarrow p=q$ or $p+q=n$
(viii) ${ }^{n} C_{r}+2{ }^{n} C_{r-1}+{ }^{n} C_{r-2}={ }^{n+2} C_{r}$ (ix) $r^{n} C_{r}=n^{n-1} C_{r-1}$
(x) The product of $k$ consecutive integers is divisible by $k$ !
Solution: Question 1(vi-x)

Question 2. Find $n$, if:
(i) ${ }^{n} \mathrm{C}_{5}={ }^{n} \mathrm{C}_{8}$ (ii) ${ }^{n} \mathrm{C}_{15}={ }^{n} \mathrm{C}_{7}$ (iii) ${ }^{n} \mathrm{C}_{50}={ }^{n} \mathrm{C}_{1}$
(iv) ${ }^{2 n} \mathrm{C}_{3}:{ }^{n} \mathrm{C}_{3}=11: 1$ (v) ${ }^{\mathrm{n}} \mathrm{C}_{6}:{ }^{\mathrm{n}-3} \mathrm{C}_{3}=33: 4$ (vi) ${ }^{2 n} C_{3}:{ }^{n} C_{2}=12: 1$
Solution: Question 2

Question 3. Find $r$, if:
(i) ${ }^{15} \mathrm{C}_{3 \mathrm{r}}={ }^{15} \mathrm{C}_{\mathrm{r}+3}$ (ii) ${ }^{8} \mathrm{C}_{\mathrm{r}}-{ }^{7} \mathrm{C}_{3}={ }^{7} \mathrm{C}_{2}$
(iii) ${ }^{16} \mathrm{C}_{\mathrm{r}}={ }^{16} \mathrm{C}_{r+4}$ (iv) ${ }^{15} \mathrm{C}_{\mathrm{r}}:{ }^{15} \mathrm{C}_{\mathrm{r}-1}=11: 5$
Solution: Question 3

Question 4. Find $n$ and $r$, if:
(i) ${ }^{n} C_{r-1}:{ }^{n} C_{r}:{ }^{n} C_{r+1}=6: 14: 21$ (ii) ${ }^{n} C_{r-1}:{ }^{n} C_{r}:{ }^{n} C_{r+1}=3: 4: 5$
(iii) ${ }^{n+1} \mathrm{C}_{r+1}:{ }^{n} \mathrm{C}_{\mathrm{r}}:{ }^{n-1} \mathrm{C}_{r-1}=22: 12: 6$ (iv) ${ }^{n} C_{r}:{ }^{n} C_{r+1}:{ }^{n} C_{r+2}=1: 2: 3$
Solution: Question 4

Question 5. In how many ways can 11 players be chosen out of 16 if
(i) there is no restriction. (ii) a particular player is always chosen.
Solution: Question 5 & 6

Question 6. Out of 5 men and 3 women, a committee of 3 is to be formed.
In how many ways can it be formed if at least one man is selected?
Solution: Question 5 & 6

Question 7. A committee of 5 members is to be formed out of 6 men and 4 women. In how many ways can it be done
if it has (i) exactly 2 women (ii) at least 2 women (iii) at most 2 women?
Solution: Question 7 & 8

Question 8. There are 10 points on a circle. Find the number of $(\mathrm{i})$ lines (ii) triangles that can be drawn?
Solution: Question 7 & 8

Question 9. Find the number of diagonals in n sided polygon?
Solution: Question 9 & 10

Question 10. In how many ways a group of 10 girls can be divided into two groups of 3 and 7 girls.
Solution: Question 9 & 10

Question 11. Number of diagonals in $n$-sided polygon is 35 . Find the number $n$ ?
Solution: Question 11 & 12

Question 12. For the post of 6 officers, there are 100 applicants, 2 posts are reserved for serving candidates and remaining for others.
Thereare 20 serving candidates among the applicants. In how many ways this selection can be made?
Solution: Question 11 & 12

Question 13. In an examination, a candidate has to pass in each of 6 subjects. In how many ways he cannot qualify the examination?
Solution: Question 13 & 14

Question 14. A question paper has three parts $\mathrm{A}, \mathrm{B}$ and C each containing 8 questions. If a student has to choose 5 questions from A ,
and 3 questions each from B and C . In how many ways can he choose the questions?
Solution: Question 13 & 14