Exercise 6.2 (Solutions)

The solutions of the Exercise 6.1 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.

The book misses the definition of the factorial function, which is defined as follows:

Question 1. Prove the following for $n \in \mathbb{N}$.
(i) ${ }^n P_r=\frac{n!}{(n-r)!}$ (ii) $\quad{ }^n P_n={ }^n P_{n-1}$ (iii) ${ }^n P_r=n^{n-l} P_{r-l}$ (iv) ${ }^n P_r={ }^{n-l} P_r+r .^{n-l} P_{r-l}$ (v) ${ }^n P_n=2 \cdot{ }^n P_{n-2}$
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) $16^n P_3=13^{n+l} P_3$ (iv) ${ }^n P_5=20{ }^n P_3$ (v) $\quad 30^n P_6={ }^{n+2} P_7$
(vi) ${ }^n P_5:{ }^{n-1} P_4=6: 1$ (vii) ${ }^n P_4:{ }^{n-l} P_3=9: 1$ (viii) ${ }^{n-1} P_3:{ }^{n+1} P_3=5: 12$ (ix) ${ }^{2 n-l} P_n:{ }^{2 n+1} P_{n-1}=22: 7$
Solution: Question 2

Question 3. Find $r$, if:
(i). ${ }^{6} P_{r-I}={ }^{5} P_{4}$ (ii). ${ }^{10} P_{r}=2{ }^{9} P_{r}$ (iii). ${ }^{15} P_{r}=210$ (iv). ${ }^{10} P_{r}=3{ }^{10} P_{r-1}$
(v). $4{ }^{6} P_{r}={ }^{6} P_{r+1}$ (vi). $\quad 2^{6} P_{r-l}={ }^{5} P_{r}$ (vii). ${ }^{54} P_{r+3}:{ }^{56} P_{r+6}=1: 30800$
Solution: Question 3

Question 4. How many 3 -digit even numbers can be formed from the digits $1,2,3,4,5,6$, if the digits are not repeated?
Solution: Question 4 & 5

Question 5. How many 7 -digits mobile number can be made using the digits 0 to 9 , if each number starts with 5 and no digit is repeated?
Solution: Question 4 & 5

Question 6. How many 4 -digit numbers can be formed with the digits $1,2,3,4,5,6$ when the repetition of the digits is allowed?
Solution: Question 6 & 7

Question 7. How many numbers can be formed with the digits $1,1,2,2,3,3,4$ so that the even digits always occupy the even places,
using all the digits and no digit is repeated?
Solution: Question 6 & 7

Question 8. In how many ways can a party of 4 men and 5 women be seated at a round table so that no two women are adjacent?
Solution: Question 8 & 9

Question 9. How many different signals can be made with 2 blue, 3 yellow and 4 green flags using all at a time.
Solution: Question 8 & 9

Question 10. How many words can be formed from the letters of the word FRIDAY? How many of them will end with F?
Solution: Question 10 & 11

Question 11. How many different permutations of the word STATESMAN can be formed using all letters at a time?
Solution: Question 10 & 11

Question 12. Find the number of arrangement of letters of the word VOWEL in which vowels may occupy odd places?
Solution: Question 12 & 13

Question 13. In how many ways can letters of word MACHINE be arranged so that all the vowels are never together?
Solution: Question 12 & 13

Question 14. How many 3 letter words (with or without meaning) can be formed out of the letter of the word ENGLISH,
if the repetition of the letter is not allowed.
Solution: Question 14 & 15

Question 15. Fatima wants to arrange 5 Mathematics, 3 English and 2 Urdu books on book shelf.
If the books on the same subjects are together, find all possible arrangements.
Solution: Question 14 & 15

Question 16. How many odd numbers can be formed by using the digits $1,2,3,4,5,6$ when repetition of digits is not allowed.
Solution: Question 16 & 17

Question 17. How many 4-digit odd numbers can be formed using the digits $1,2,3,4$ and 5 if no digit is repeated.
Solution: Question 16 & 17

Question 18. How many odd numbers less than 10,000 can be formed using the digits $0,2,3,5,6$ without repeating the digits.
Solution: Question 18 & 19

Question 19. The chief secretary of Sindh calls a meeting of 10 secretaries. In how many ways they be seated at a round table
if three particular secretaries want to sit together?
Solution: Question 18 & 19

Question 20. Find the number of ways that 6 men and 6 women seated at a round table such that they occupy alternative seats.
Solution: Question 20 & 21

Question 21. Make all the permutations of the following words WHY, SAD, TWO, MADE
Solution: Question 20 & 21

Question 22. Encrypt the word LAHORE by using the permutation: $$ \left(\begin{array}{llllll} 3 & 4 & 6 & 1 & 5 & 2 \end{array}\right) $$

By labelling L as $1, \mathrm{~A}$ as 2 and so on.
Solution: Question 22 & 23

Question 23. Decrypt the word “TNLUMA” by using the permutation:

$$ \left(\begin{array}{llllll} 4 & 6 & 3 & 2 & 1 & 5 \end{array}\right). $$
Solution: Question 22 & 23