ALLENHOUSE PUBLIC SCHOOL, JHANSI

CLASS: XI: MATHEMATICS

PERMUTATIONS AND COMBINATIONS
PR A CTICE QUESTIONS ON F A CTORI A L A ND
FUND A MENT A L PRINCIPLES OF COUNTING
FORMULA USED

Factorial notation: n ! or n
1. n !  n(n  1)n  2)(n  3)........3.2.1
2. 0!  1
When n is negative or a fraction, n! is not defined.

Prove the following for n  N ,

1. (2n)!  2n.n !.[1.3.5.......(2n  1)]
2. (n  1)[n !n  (n  1)!(2n  1)  (n  2)!(n  1)!]  (n  2)!
n! n! (n  1)!
3.  
r ! n  r  ! (r  1)! n  r  1! r ! n  r  1!

n!
4.  n(n  1)(n  2).....(r  1)
r!
n! n!
5. (n  r  1). 
(n  r  1)! (n  r )!

6. 33! is divisible by 215

(2n !) n(n  1)(n  2)....(2n  1)(2n)
7. 
[(n  1)!]2
(n  1)!
8. (n ! 1) is not divisible by any natural number between 2 and n.

9. (n !)2  n n .n !  (2n)!
Find n, if
n!
10.  930, n  2
(n  2)!
n! n!
11.  20. ,n  5
(n  5)! (n  3)!
12. (n  2)!  60.(n  1)!
13. (n  2)!  2550.n !
1 1 n
14.  
9! 10! 11!
n!
15.  990
(n  3)!

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 (n  1)!
16. 6
(n  1)!
(n  2)! (2n  1)! 72
17. . 
(2n  1)! (n  3)! 7
(2n)! n!
18. :  52 : 5
5!(2n  3)! 4!(n  2)!
n! n!
19. :  2 :1, n  4
2(n  2)! 4!(n  4)!
20. How many 3 letter words(with or without meaning) can be formed out of the letters of the word
LOGARITHMS, if repetition of letter is not allowed?
21. Aditi wants to arrange 4 English, 3 Maths and 2 Physics books on a shelf. If the books on the
same subject are different, determine the number of all possible arrangement.
22. A flag is in the form of three blocks each to be coloured differently. If there are eight different
colours to choose from, how many such flags are possible?
23. In how many ways two books of different languages can be selected from 10 Hindi, 5 English
and 7 Sanskrit books?
24. Four persons enter a bus and they find seven seats vacant. In how many ways can they be seated?
25. How many 5-digit numbers are there with all distinct digits?
26. How many digit odd numbers can be formed by using the digits 1,2,3,4,5,6 when (i) repetition of
digits is not allowed (ii) the repetition of digits is allowed?
27. How many words are there with or without meaning of three distinct alphabets?
28. There are four routes between Delhi and Mumbai. In how many ways can a person go from
Delhi to Mumbai and return if for returning (i) any route is taken (ii) the same route is taken (iii)
the same route is not taken.
29. How many 4-digits odd numbers can be formed with the help of the digits 1,2,3, 4 and 5 if (i) no
digit is repeated (iii) digits are repeated?
30. How many odd numbers less than 10,000 can be formed using the digits 0, 2, 3, 5 allowing
repetition of digits?
31. How many 4-digits numbers can be formed using the digits 0,1,2,3,4,5, no digit being repeated?
32. How many 3-digit numbers are there such that 5 is at units place?
33. How many numbers are there between 100 and 1000 such that at least one of the digits is 6?
34. How many then three digit numbers are there which have exactly one of the digits as 6?
35. For a set of six true or false questions, no student has written all answers and no two students
have given the same sequence of answers. What is the maximum number of students in the class
for this job to be possible?

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 ALLENHOUSE PUBLIC SCHOOL, JHANSI
CLASS: XI: MATHEMATICS
PERMUTATIONS AND COMBINATIONS
PR A CTICE QUESTIONS ON PERMUT A TIONS
FORMULA USED

The different arrangement which can be made out of a given number of things by
taking some or all at a time, are called permutations.
Let 1  r  n , then the number of all permutations of n dissimilar things taken at a time
is given by n Pr or P (n, r )
n!
n
Pr  n(n  1)(n  2)......(n  r  1) 
(n  r )!
Properties: Pn  n !,
n n
Pn1  n !, n
P0  1
Circular Permutation:
The number of circular permutation of n different objects is (n – 1)!
The number of ways in which n persons can be seated round a table is (n – 1)!
The number of ways in which n different beads can be arranged to form a necklace is
1
(n  1)!
2

Prove the following for n  N ,

n!
1. n
Pr 
(n  r )!

2. n
Pn  n Pn 1

3. n
Pr  n. n1Pr 1
n 1
4. n
Pr  Pr  r . n1Pr 1

5. n
Pn  2. n Pn 2

Find n, if,
6. n
P4  20 n P2

7. 2n
P3  100. n P2

8. 16. n P3  13. n1P3

9. n
P5  20. n P3
n 2
10. 30. n P6  P7

11. n P5 : n1P4  6 :1

12. n P4 : n1P3  9 :1
n 1
13. P3 : n 1P3  5 :12

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 2 n 1
14. Pn : 2 n1Pn 1  22 : 7

Find r, if
15. 5 P4  6 Pr 1

16. 15 Pr  2730

17. 10 Pr  2 9 Pr

18. 4. 6 Pr  6 Pr 1

19. 20
Pr  13. 20 Pr 1

20. 5 Pr  2. 6 Pr 1

21. 56
Pr  6 : 54 Pr  3  30800 :1

22. How many three-digit even numbers can be formed from the digits 1,2,3,4,5,6 if the digits can be
repeated?
23. How many 5 digit telephone number can be made using the digits 0 to 9, if each number starts
with 67 and no digit appears more than once?
24. In how many ways can a party of 4 men and 4 women be seated at a circular table so that no two
women are adjacent?
25. 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?
26. How many numbers can be formed with the digits 1,2,3,4,3,2,1 so that the odd digits always
occupy the odd places?
27. How many different signals can be made from 4 red, 2 white and 3 green flags by arranging all
of them vertically on a flag staff?
28. There are how many types of calendar for the month of February?
29. In how many ways can 4 letters be posted in 3 letter boxes?
30. A boy has 6 pockets. In how many ways can he put 5 coins in his pockets?
31. In how many ways can three prizes be distributed among 4 boys when (i) no one gets more than
one prize (ii) a boy can get any number of prizes.
32. How many different permutations each containing the letter of the word STATESMAN can be
formed?
33. Find the number of ways in which the letter of the word MACHINE can be arranged such that
the vowels may occupy only odd ppositions.
34. How many words can be formed from the letters of the word SUNDAY? How many of these
begin with D?
35. In how many ways can be letters of the word DIRECTOR be arranged so that all the vowels are
never together?

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 CLASS: XI: MATHEMATICS
PERMUTATIONS AND COMBINATIONS
PR A CTICE QUESTIONS ON COMBIN A TIONS
FORMULA USED

Each of the different groups or selections which can be formed by taking some or all of
number of objects, irrespective of their arrangement, is called a combination.
The number of all combinations of n distinct objects, taken r at a time is given by
n
Cr or C(n, r ) . nCr is defined only when n and r are integers such that
n  r , n  0 and r  0
n!
n
Cr 
r !(n  r )!
Properties:
n
Cr  nCn r
n 1
n
Cr  nCr 1  Cr
n
C0  1, Cn  1
n

If nC p  nCq , then p = q or + q = n.

Prove the following for n  N ,

n!
1. n
Cr 
r !(n  r )!

2. n. n 1Cr 1  (n  r  1) nCr 1
n
Cr n  r 1
3. n

Cr 1 r
n 1
4. Cr 1  n1Cr  nCr
n 1
5. n
Cr  nCr 1  Cr

2n.[1.3.5.....(2n  1)]
6. 2n
Cn 
n!
7. n
C p  nCq  p  q or p  q  n
n2
8. n
Cr  2. nCr 1  nCr  2  Cr
n
Cr n
9. n 1

Cr 1 r
10. The product of k consecutive positive integers is divisible by k!.
Find n, if
11. nC7  nC5

12. nC10  nC15

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13. nC30  nC4

14. 2n
C3 : nC3  11:1

15. nC6 : n 3C3  33 : 4

16. 2n
C3 : nC2  12 :1

Find r, if
17. 15C3r  15Cr  3

18. 8Cr  7C3  7C2

19. 18Cr  18Cr  2

20. 15Cr : 15Cr 1  11: 5

Find n and r, if
21. nCr 1  36, nCr  84, nCr 1  126

22. nCr 1 : nCr : nCr 1  3 : 4 : 5

23. n Pr  n Pr 1 , nCr  nCr 1

n 1
24. Cr 1 : nCr  11: 6, nCr : n1Cr 1  2 :1

25. nCr : nCr 1 : nCr  2  1: 2 : 3

26. In how many ways can 11 players be chosen our of 15 if (i) there is no restriction (ii) a particular
player is always be chosen (iii) a particular player is never chosen?
27. Out of 5 men and 2 women, a committee of 3 is to be formed. In how many ways can it be
formed if at least one woman is to be included?
28. A committee of 5 is to be formed out of 6 men and 4 women. In how many ways can this be
done, if (i) at least 2 women are included (ii) at most 2 women are included?
29. There are n points on a circle, find the number of (i) lines which can be drawn (ii) triangles
which can be formed.
30. How many diagonals are there in a polygon of n sides?
31. A polygon has 35 diagonals. Find the number of its sides.
32. In how many ways a group of 11 boys can be divided into two groups of 6 and 5 boys each?
33. For the post of 5 clerks, there are 25 applicants, 2 posts are reserved for SC candidates and
remaining for others, there are 7 SC candidates among the applicants. In how many ways can the
selection be made?
34. In how many ways can 10 different books on English and 5 similar books on Hindi be placed in
a row on a shelf so that two books on Hindi are not together?
35. In an examination, a candidate has to pass in each of the 5 subjects. In how many ways can he
fail?
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