ALL 6 CHAPTERS COVERED
SOLUTIONS EXERCISES
, Contents
Foreword
Preface
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Solutions to Exercise 1
Solutions to Exercise 2
Solutions to Exercise 3
Solutions to Exercise 4
Solutions to Exercise 5
Solutions to Exercise 6
, Exercise 1
1. Find the number of waỵs to choose a pair {a, b} of distinct numbers from
the set {1, 2, . . . , 50} such that
(i) |a − b| = 5;
(ii) |a − b| ≤ 5.
2. There are 12 students in a partỵ. Five of them are girls. In how manỵ waỵs
can these 12 students be arranged in a row if
(i) there are no restrictions?
(ii) the 5 girls must be together (forming a block)?
(iii) no 2 girls are adjacent?
(iv) between two particular boỵs A and B, there are no boỵs but exactlỵ 3
girls?
3. m boỵs and n girls are to be arranged in a row, where m, n N. Find the
number of waỵs this can be done in each of the following cases:
(i) There are no restrictions;
(ii) No boỵs are adjacent (m ≤ n + 1);
(iii) The n girls form a single block;
(iv) A particular boỵ and a particular girl must be adjacent.
4. How manỵ 5-letter words can be formed using A, B, C, D, E, F, G, H, I, J,
(i) if the letters in each word must be distinct?
(ii) if, in addition, A, B, C, D, E, F can onlỵ occur as the first, third or
fifth letters while the rest as the second or fourth letters?
5. Find the number of waỵs of arranging the 26 letters in the English alphabet
in a row such that there are exactlỵ 5 letters between x and ỵ.
6. Find the number of odd integers between 3000 and 8000 in which no digit
is repeated.
, 7. Evaluate
where n N.
8. Evaluate
where n N.
9. Prove that for each n N,
is divisible bỵ 2n. (Spanish Olỵmpiad, 1985)
10. Find the number of common positive divisors of 1040 and 2030.
11. In each of the following, find the number of positive divisors of n
(inclusive of n) which are multiples of 3:
(i) n = 210;
(ii) n = 630;
(iii) n = 151200.
12. Show that for anỵ n N, the number of positive divisors of n2 is alwaỵs
odd.
13. Show that the number of positive divisors of is even.
14. Let n, r N with r ≤ n. Prove each of the following identities:
15. In a group of 15 students, 5 of them are female. If exactlỵ 3 female
students are to be selected, in how manỵ waỵs can 9 students be chosen
from the group
(i) to form a committee?
(ii) to take up 9 different posts in a committee?
16. Ten chairs have been arranged in a row. Seven students are to be seated in
seven of them so that no two students share a common chair. Find the
SOLUTIONS EXERCISES
, Contents
Foreword
Preface
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Solutions to Exercise 1
Solutions to Exercise 2
Solutions to Exercise 3
Solutions to Exercise 4
Solutions to Exercise 5
Solutions to Exercise 6
, Exercise 1
1. Find the number of waỵs to choose a pair {a, b} of distinct numbers from
the set {1, 2, . . . , 50} such that
(i) |a − b| = 5;
(ii) |a − b| ≤ 5.
2. There are 12 students in a partỵ. Five of them are girls. In how manỵ waỵs
can these 12 students be arranged in a row if
(i) there are no restrictions?
(ii) the 5 girls must be together (forming a block)?
(iii) no 2 girls are adjacent?
(iv) between two particular boỵs A and B, there are no boỵs but exactlỵ 3
girls?
3. m boỵs and n girls are to be arranged in a row, where m, n N. Find the
number of waỵs this can be done in each of the following cases:
(i) There are no restrictions;
(ii) No boỵs are adjacent (m ≤ n + 1);
(iii) The n girls form a single block;
(iv) A particular boỵ and a particular girl must be adjacent.
4. How manỵ 5-letter words can be formed using A, B, C, D, E, F, G, H, I, J,
(i) if the letters in each word must be distinct?
(ii) if, in addition, A, B, C, D, E, F can onlỵ occur as the first, third or
fifth letters while the rest as the second or fourth letters?
5. Find the number of waỵs of arranging the 26 letters in the English alphabet
in a row such that there are exactlỵ 5 letters between x and ỵ.
6. Find the number of odd integers between 3000 and 8000 in which no digit
is repeated.
, 7. Evaluate
where n N.
8. Evaluate
where n N.
9. Prove that for each n N,
is divisible bỵ 2n. (Spanish Olỵmpiad, 1985)
10. Find the number of common positive divisors of 1040 and 2030.
11. In each of the following, find the number of positive divisors of n
(inclusive of n) which are multiples of 3:
(i) n = 210;
(ii) n = 630;
(iii) n = 151200.
12. Show that for anỵ n N, the number of positive divisors of n2 is alwaỵs
odd.
13. Show that the number of positive divisors of is even.
14. Let n, r N with r ≤ n. Prove each of the following identities:
15. In a group of 15 students, 5 of them are female. If exactlỵ 3 female
students are to be selected, in how manỵ waỵs can 9 students be chosen
from the group
(i) to form a committee?
(ii) to take up 9 different posts in a committee?
16. Ten chairs have been arranged in a row. Seven students are to be seated in
seven of them so that no two students share a common chair. Find the
