˜ 1. When
a job (task) is performed in different ways then each way is called Permutation.
˜ 2. Fundamental
Principle of Multiplication: If a job can be performed in m different ways and for
each such way, second job can be done in n different ways, then the two jobs
(in order) can be completed in m × n ways.
˜ 3. Fundamental
Principle of Addition: If there are two events such that they can be performed
independently in m and n ways respectively, then either of the two events can
be performed in (m + n) ways.
Difference between permutations and combinations:-
˜
VERY SHORT ANSWER TYPE QUESTIONS
1. Using the digits 1, 2, 3, 4, 5 how many 3 digit numbers
(without repeating the digits) can be made?
2. In how many ways 7 pictures can be hanged on 9 pegs?
3. Ten buses are plying between two places A and B. In how
many ways a person can travel from A to B and come back?
4. There are 10 points on a circle. By joining them how
many chords can be drawn?
5. There are 10 non collinear points in a plane. By
joining them how many triangles can be made?
8.
How many different words (with or without meaning) can
be made using all the vowels at a time?
9. Using 1, 2, 3, 4, 5 how many numbers greater than 10000 can be made? ( Repetition not allowed )
10. If nC12 = nC13 then find the value of 25Cn.
11. In how many ways 4 boys can be chosen from 7 boys to make a committee?
12. How many different words can be formed by using all the letters of word SCHOOL?
13. In how many ways can the letters of the word PENCIL be arranged so that I is always next to L
9. Using 1, 2, 3, 4, 5 how many numbers greater than 10000 can be made? ( Repetition not allowed )
10. If nC12 = nC13 then find the value of 25Cn.
11. In how many ways 4 boys can be chosen from 7 boys to make a committee?
12. How many different words can be formed by using all the letters of word SCHOOL?
13. In how many ways can the letters of the word PENCIL be arranged so that I is always next to L
SHORT ANSWER TYPE QUESTIONS
14.
In how many ways 12 boys can be seated on 10 chairs in
a row so that two particular boys always take seat?
15. In
how many ways 7 positive and 5 negative signs can be arranged in a row so that
no two negative signs occur together?
16. From
a group of 7 boys and 5 girls, a team consisting of 4 boys and 2 girls is to be
made. In how many different ways it can be done?
17. In
how many ways can one select a cricket team of eleven players from 17 players
in which only 6 players can bowl and exactly 5 bowlers are to be included in
the team?
18. In
how many ways 11 players can be chosen from 16 players so that 2 particular
players are always excluded?
19. Using
the digits 0, 1, 2, 2, 3 how many numbers greater than 20000 can be made?
20. If
the letters of the word ‘PRANAV’ are arranged as in dictionary in all possible
ways, then what will be 182nd word
21. From
a class of 15 students, 10 are to chosen for a picnic. There are two students
who decide that either both will join or none of them will join. In how many
ways can the picnic be organized?
22. Using
the letters of the word, ‘ARRANGEMENT’ how many different words (using all
letters at a time) can be made such that both A, both E, both R and both N
occur together.
23. A
polygon has 35 diagonals. Find the number of its sides.
[Hint: Number of diagonals of n
sided polygon is given by nC2 – n]
24.
How many different products can be obtained by
multiplying two or more of the numbers 2, 3, 6, 7, 9?
25. Determine
the number of 5 cards combinations out of a pack of 52 cards if at least 3 out
of 5 cards are ace cards?
26.
How many words can be formed from the letters of the
word ‘ORDINATE’ so that vowels occupy odd places?
LONG ANSWER TYPE QUESTION
27.
Using the digits 0, 1, 2, 3, 4, 5, 6 how many 4 digit
even numbers can be made, no digit being repeated?
28. There
are 15 points in a plane out of which 6 are in a straight line, then (a) How
many different straight lines can be made?
(b) How
many triangles can be made?
(c) How
many quadrilaterals can be made?
29. If
there are 7 boys and 5 girls in a class, then in how many ways they can be
seated in a row such that (i) No two girls sit together?
(ii) All the girls never sit together?
30. Using
the letters of the word 'EDUCATION' how many words using 6 letters can be made
so that every word contains at least 4 vowels?
31. What
is the number of ways of choosing 4 cards from a deck of 52 cards? In how many
of these,
(a)
3 are red and 1 is black.
(b)
All 4 cards are from different suits.
(c)
At least 3 are face cards.
(d)
All 4 cards are of the same colour.
32. How
many 3 letter words can be formed using the letters of the word INEFFECTIVE?
33.
How many 5 letter words containing 3 vowels and 2
consonants can be formed using the letters of the word EQUATION so that 3
vowels always occur together?
34.
Using the digits 1, 2, 3, 4 how many 4 digit natural
numbers less than 4321 can be made, if digits can repeated?
35.
Using the digits 1, 2, 3, 4 how many natural numbers
less than 4321 can be made, if digits can repeated?
ANSWERS
1. 60 3.
100 4.
45 5. 120 6. 513
7. n = 6 8. 120 9.
120 10. 1 11. 35 12. 360
13. 120 14. 90 × 10P8 15. 56 16. 350 17. 2772 18.
364
19. 36 20. PAANVR 21. 13C10 + 13C8 22. 5040 23. 10
24. 26 25. 4560 26. 576 27. 420
28. (a) 91
(b) 435
(c)
Hint: For
quadrilaterals selection of 4 points can be done like this,
9C4 + 9C2 × 6C2 + 9C3 × 6C1
29. (i)
7! × 8P5 (ii)
12! – 8! × 5! 30. 24480
31. 52C4
(a)
26C1 × 26C3
(b)
(13)4
(c)
9295 ( Hint : Face cards : 4J + 4K + 4Q )
(d)
2 × 26C4
32. 265 [Hint:
make 3 cases i.e.
(i)
All 3 letters are different
(ii) 2 are identical 1 different
(iii)
All are identical, then form the words.)
33. 1080 34. 229, 35.
313
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