# 65. Permutations and Combinations Problems Part 3

1. Find the number of permutations of the letters of the word “Independence”.

2. How many different words can be formed with the letters of the word “Principal” so that the vowels are together?

3. How many words can be formed with the letters of the word “Mathematics”? In how many of them the vowels are together and consonants are together?

4. In how many ways can the letters of the word “Director” be arranged so that the three vowels are together?

5. In how many ways can the letters of word “Plantain” be arranged so that the three vowels are together?

6. Find the number of words that can be made by arranging the letters of the word “Intermediate” so that the relative order of vowels and consonants do not change.

7. In how many permutations of the word “Parallel” all the $$l$$ do not come together?

8. Find the number of words formed by the letters of the word ‘Delhi’ which

1. begin with D

2. end with I

3. the letter L being always in the middle

4. begin with D and end with I

9. In how many ways can the letters of the word ‘violent’ be arranged so that vowels occupy only the odd places?

10. In how many ways can the letters of the word ‘Saloon’ be arranged if the consonants and vowels must occupy alternate places?

11. How many words can be formed out of the word “Article” so that vowels occupy the even places?

12. How many numbers greater than four million can be formed with the digits $$2, 3, 0, 3, 4, 5, 5$$?

13. How many seven digit numbers can be formed with the digits $$1, 2, 2, 2, 3, 3, 5$$? How many of them are odd?

14. How many numbers of seven digits can be formed with the digits $$1, 2, 3, 4, 3, 2, 1,$$ so that odd digits always occupy the odd places?

15. How many number greater than $$10,000$$ can be formed with the digits $$1, 1, 2, 3, 4, 0$$?

16. Find the number of numbers of four digits that can be made from the digits $$0, 1, 2, 3, 4, 5$$ if the digits can be repeated in the same number. How many of these numbers have at least one digit repeated?

17. How many signals can be made by hoisting $$2$$ blue, $$2$$ red and $$5$$ yellow flags on a pole at the same time?

18. How many different signals can be made by hoisting $$6$$ differently colored flags one above the other when any number of them may be hoisted at once?

19. Find the number of arrangements of the letter of the word “Delhi” if $$e$$ always comes before $$i$$.

20. In how many ways can $$5$$ men sit around a table?

21. In how many ways $$5$$ boys and $$5$$ girls can sit around a table?

1. If there is no restriction

2. If no two girls sit side by side.

22. In a class of students there are $$6$$ boys and $$4$$ girls. In how many ways can they be seated around a table so that all $$4$$ girls sit together?

23. $$5$$ boys and $$5$$ girls form a line with the buys and girls alternating. Find the number of ways in which line can be made. In how many different ways could they form a circle so that boys and girls alternate?

24. In how many ways $$6$$ boys and $$5$$ girls can sit at a round table when no two girls sit next to each other?

25. In how many ways $$50$$ pearls be arranged to form a necklace?

26. A round table conference is to be held between $$20$$ delegates of $$20$$ countries. In how many ways they and the and the host can be seated if two particular delegated are always to sit on either side of the host?

27. Four gentlemen and four ladies are invited to a certain party. Find the number of ways of seating them around a table so that only ladies are seated on the two sides of each gentleman.

28. In how many ways can $$7$$ Englishman and $$6$$ Indians sit around a table so that no two Indians are together?

29. If $$^{15}C_{3r}=^{15}C_{r + 3}$$, find $$r$$.

30. Find $$n$$, if $$^nC_6:^{n - 3}C_3 = 33:4$$

31. Find the value of the expression $$^{47}C_4 + \sum_{j=1}^5~^{52-j}C_3$$

32. Prove that the product of $$r$$ consecutive integers is divisible by $$r!$$

33. Find the number of triangles which can be formed by joining the angular points of a polygon of $$m$$ sides as vertices.

34. A man has $$8$$ children to take them to a zoo. He takes three of them at a time to the zoo as often as he can without the same $$3$$ children together more than once. How many times will he have to go to zoo? How many times a particular child will go?

35. On a new year day every student of a class sends a card to every other student. The postman delivers $$600$$ cards. How many students are there in the class.

36. Show that a polygon of $$m$$ sides has $$\frac{m(m - 3)}{2}$$ diagonals.

37. Out of $$6$$ gentlemen and:math:4 ladies a committee of $$5$$ is to be formed. In how many ways can this be done so as to include at least one lady in each committee?

38. There are ten points in a plane. Of these ten points four points are in a straight line. With the exception of these four points no other three points are in the same straight line. Find

1. the number of triangles formed.

2. the number of straight lines formed

3. the number of quadrilaterals formed, by joining these ten points.

39. There are $$4$$ oranges, $$5$$ apples and $$6$$ mangoes in a fruit basket. In how many ways a person make a selection of fruits from the fruits in the basket.

40. Given $$5$$ different green dyes, $$4$$ different blue dyes and $$3$$ different red dyes. How many combinations of dyes can be chosen taking at least one green and one blue dye.

41. Find the number of divisors of $$216000$$.

42. In an examination a minimum is to be secured in each of $$5$$ subjects to pass. In how many ways can a student fail?

43. In how many ways $$12$$ different things can be divided equally among $$3$$ persons.

44. How many different words of $$4$$ letters can be formed with the letters of the word “EXAMINATION”?

45. If $$^nC_{30} =~^nC_4$$, find $$n$$.

46. If $$^nC_{12} =~^nC_8$$, find $$^nC_{17}$$ and $$^{22}C_n$$.

47. If $$^{18}C_r =~^{18}C_{r+2}$$, find $$^rC_6$$.

48. If $$^nC_{n - 4}= 15$$, find $$n$$.

49. If $$^{15}C_r:~^{15}C_{r - 1} = 11:5$$, find $$r$$.

50. If $$^nP_r = 2520$$ and $$^nC_r = 21$$, find $$r$$.