Permutations and Combinations

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

2. Find the value of (i) 60P

_{3}(ii)4P_{4 }Ans: 205320, 243. Find the value of (i) 10C

_{3}(ii) 100C_{98}(iii) 50C_{50}Ans: 120, 4950, 14. 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)