Permutations and Combinations

1.
Evaluate: 30! /28! Ans: 870

2.
Find the value of
(i) 60P

_{3}(ii)4P_{4 }Ans: 205320, 24
3.
Find the value of
(i) 10C

_{3}(ii) 100C_{98}(iii) 50C_{50}Ans: 120, 4950, 1
4.
How many words
can be formed by using all letters of the word “BIHAR”? Ans: 120

5.
How many words
can be formed by using all letters of the word ‘DAUGHTER’ so that the vowels
always come together? Ans:
4320

6.
How many words
can be formed from the letters of the word ‘EXTRA’ so that the vowels are never
together? Ans: 72

7.
How many words
can be formed from the letters of the word ‘DIRECTOR’ So that the vowels are
always together? Ans:
2160.

8.
In how many ways
can a cricket eleven be chosen out of a batch of 15 players? (1365)

9.
In how many ways,
a committee of 5 members can be selected from 6 men and 5 ladies, consisting of
3 men and 2 ladies? Ans: 200.

Exercise

1.
The value of 75P

_{2}is : (5550)
2.
How many 4-letter
words with or without meaning, can be formed out of the letters of the word,
'LOGARITHMS', if repetition of letters is not allowed? (5040)

3.
How many words
with or without meaning, can be formed by using all the letters of the word,
'DELHI', using each letter exactly once?
(120)

4.
In how many ways
can the letters of the word 'APPLE' be arranged? (60)

5.
In how many ways
can the letters of the word 'LEADER' be arranged? (360)

6.
In how many
different ways can the letters of the word 'RUMOUR' be arranged? (180)

7.
How many words
can be formed by using all the letters of the word, 'ALLAHABAD'?(7560)

8.
How many
arrangements can be made out of the letters of the word 'ENGINEERING'?(277200)

9.
How many words
can be formed from the letters of the word 'SIGNATURE' so that the vowels
always come together ?(17280)

10.
In how many
different ways can the letters of the word 'OPTICAL' be arranged so that the
vowels always come together?(4320)

11.
In how many different ways can the letters of
the word 'SOFTWARE' be arranged in such a way that the vowels always come
together? (720)

12.
In how many different ways can the letters of
the word 'LEADING' be arranged in such a way that the vowels always come
together?(720)

13.
In how many different ways can the letters of
the word 'JUDGE' be arranged in such a way that the vowels always come together?(48)

14.
In how many
different ways can the letters of the word 'AUCTION' be arranged in such a way
that the vowels always come together? (576)

15.
In how many
different ways can the letters of the word 'BANKING' be arranged so that the
vowels always come together?(720)

16.
In how many
different ways can the letters of the word 'CORPORATION' be arranged so that
the vowels always come together?(50400)

17.
In how many
different ways can the letters of the word 'MATHEMATICS' be arranged so that
the vowels always come together?(120960)

18.
In how many
different ways can the letters of the word 'DETAIL be arranged in such a way
that the vowels occupy only the odd positions?(36)

19.
In how many
different ways can the letters of the word 'MACHINE' be arranged so that the
vowels may occupy only the odd positions?(576)

20.
In how many ways
can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?(63)

21.
In how many ways a committee, consisting of 5
men and 6 women can be formed from 8 men and 10 women?(11760)

22.
From a group of 7 men and 6 women, five
persons are to be selected to form a committee so that at least 3 men are there
on the committee. In how many ways can it be done?(756)

23.
In a group of 6
boys and 4 girls, four children are to be selected. In how many different ways
can they be selected such that at least one boy should be there?(209)

24.
A box contains 2 white balls, 3 black balls
and 4 red balls. In how many ways can 3 balls be drawn from the box, if at
least one black ball is to be included in the draw?(64)

25.
How many 3-digit numbers can be formed from
the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits
is repeated?(20)

26.
In how many ways
can 21 books on English and 19 books on Hindi be placed in a row on a shelf so
that two books on Hindi may not be together?(1540)

27.
Out of 7
consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be
formed? (25200)