# 67. Permutations and Combinations Problems Part 4

1. Prove that $$^{20}C_{13} + ^{20}C_{14} - ^{20}C_6 -^{20}C_7 = 0$$

2. If $$^nC_{r - 1} = 36, ^nC_r = 84$$ and $$^nC_{r+1} = 126$$, find $$n$$ and $$r$$.

3. How many quadrilaterals can be formed by joining vertices of a polygon of $$n$$ sides.

4. A man has $$7$$ friends and he wants to invite $$3$$ of them at a party. Find out how many parties to each of his $$3$$ different friends he can give and how many times any particular friend will attend the parties.

5. A delegation of $$6$$ members is to be sent abroad out of $$12$$ members. In how many ways can the selection be made so that

1. a particular member is always included.

2. a particular member is always excluded.

6. There are six students A, B, C, D, E, F.

1. In how many ways can they be seated in a line so that C and D do not sit together?

2. In how many ways can a committee of $$4$$ be formed so as to always include C?

3. In how many ways can a committee of $$4$$ be formed so as to always include C but exclude E?

7. There are $$n$$ stations in a railway line. The number of kinds of tickets printed(no return ticket) is $$105$$. Find the number of stations.

8. There are $$15$$ points in a plane of which $$6$$ are collinear. How many different straight lines and triangles can be drawn by joining them?

9. There are $$10$$ points in a plane out of which $$5$$ are collinear. Find the number of quadrilaterals formed having vertices at points.

10. The three sides of a triangle have $$3, 4, 5$$ interior points on them. Find the number of triangles that can be constructed using given interior points as vertices.

11. In how many ways can a team of $$11$$ be chosen from $$14$$ football players if two of them can be only goalkeepers?

12. A committee of $$2$$ men and $$2$$ women is to be chosen from $$5$$ men and $$6$$ women. In how many ways can this be done?

13. Find the number of ways in which $$8$$ different articles can be distributed among $$7$$ boys, if each boy is to receive at least one article.

14. Out of $$7$$ men and $$4$$ ladies a committee of $$5$$ is to be formed. In how many ways can this be done so as to include at least $$3$$ ladies.

15. A candidate is required to answer six out of ten questions which are divided into two groups, each containing five questions and he is not permitted to attempt more than $$4$$ from any group. In how many ways can he make up his choices?

16. There are $$10$$ professors and $$20$$ students out of whom a committee of $$2$$ professors and $$3$$ students is to be formed. Find in how many ways these committees can be formed if

1. a particular professor is included?

2. a particular professor is excluded?

17. From $$6$$ boys and $$7$$ girls a committee of $$5$$ is to be formed so as to include at least one girl. Find the number of ways in which this can be done.

18. From $$6$$ gentlemen and $$4$$ ladies, a committee of $$5$$ is to be formed. In how many ways can this be done if

1. there is no restriction?

2. the committee is to include at least one lady?

19. From $$8$$ gentlemen and $$4$$ 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?

20. In a group of $$15$$ boys, there are $$6$$ hockey players. In how many ways can $$12$$ boys be selected so as to include at least $$4$$ hockey players?

21. From $$7$$ gentlemen and $$4$$ ladies a boat party of $$5$$ is to be formed. In how many ways can this be done so as to include at least one lady?

22. A committee of $$6$$ is to be formed out of $$4$$ boys and $$6$$ girls. In how many ways can this be done if girls may not be outnumbered?

23. A person has $$12$$ friends out of which $$8$$ are relatives. In how many ways can he invite $$7$$ friends such that at least $$5$$ of them are relatives?

24. A student is required to answer $$7$$ questions out of $$12$$ questions which are divided into two groups each containing $$6$$ questions. He is not permitted to attempt more than $$5$$ from either group. In how many ways can he choose the seven questions?

25. Each of two parallel lines has a number of distinct points marked on them. On one line there are $$2$$ points P and Q and on the other there are $$8$$ points.

1. Find the number of triangles formed having three of the $$10$$ points as vertices.

2. How many of these triangles include P but exclude Q?

26. There are $$7$$ men and $$3$$ ladies contesting for $$2$$ vacancies; an elector can vote for any number of candidates not exceeding no. of vacancies. In how many ways can he vote?

27. A party of $$6$$ is to be formed from $$10$$ boys and $$7$$ girls so as to include $$3$$ boys and $$3$$ girls. In how many ways can this party be formed if two particular girls refuse to join the same party?

28. In an examination, the question paper contains three different sections A, B and C containing $$4, 5$$ and $$6$$ questions respectively. In how many ways, a candidate can make a selection of $$7$$ questions, selecting at least $$2$$ questions from each section.

29. From $$5$$ apples, $$4$$ oranges and $$3$$ mangoes, how many selections of fruits can be made?

30. Find the total no. of selections of at least one red ball from $$4$$ red and $$3$$ green balls if the balls of same color are different.

31. Find the number of different sums that can be formed with one rupee, one half rupee and one quarter rupee coin.

32. There are $$5$$ questions in a question paper. In how many ways can a boy solve one or more questions?

33. In an election for $$3$$ seats there are $$6$$ candidates. A voter cannot vote for more than $$3$$ candidates. In how many ways can he vote?

34. In an election the number of candidates is one more than the number of members to be elected. If a voter can vote in $$30$$ different ways, find the number of candidates. (A voter has to vote for at least one candidate.)

35. In how many ways $$12$$ different books can be distributed equally among $$4$$ persons.

36. In how many ways $$10$$ mangoes can be distributed among $$4$$ persons if any person can get any number of mangoes?

37. How many words can be formed out of $$10$$ consonants and $$4$$ vowels, such that each contains $$3$$ consonants an $$2$$ vowels?

38. A table has $$7$$ seats, $$4$$ being on one side facing the window and three being on the opposite side. In how many ways can seven people be seated at the table if $$3$$ people must sit on the side facing the window.

39. A tea party is arranged for $$16$$ people along two sides of a long table with $$8$$ chairs on each side. Four men wish to sit on one particular side and two on the other side. In how many ways can they be seated?

40. Eight chairs are numbered $$1$$ to $$8$$. Two women and three men wish to occupy one chair each. First two women choose chairs from amongst the chair marked $$1$$ to $$4$$; and then men select the chairs from remaining. Find the number of possible arrangements.

41. Show that $$^{2n}C_r(0\leq r \leq 2n)$$ is greatest when $$r = n$$.

42. Ten different letters of an alphabet are given. Words with five letter are formed from these given letters. Find the number of words which have at least one letter repeated.

43. How many ternary sequences of length $$9$$ are there which either begin with $$210$$ or end with $$210$$

44. Find the number of $$7$$ digit numbers when the sum of those digits is even.

45. In how many ways $$10$$ Indians, $$5$$ Americans and $$4$$ British can be seated in a row so that all Indians are together.

46. In how many ways can the letters of the word “ARRANGE” be arranged so that

1. the two R’s are never together?

2. the two A’s are together but not the two R’s?

3. neither the two A’s nor the two R’s are together?

47. A man invites a party of $$m + n$$ friends to dinner and places $$m$$ at one round table and $$n$$ at another. Find the number of arranging the guests.

48. Find the total no. of signals that can be made by five flags of different colors when any number of them may be used.

49. The letters of the word OUGHT are written in all possible orders and these words are written out in a dictionary. Find the rank of the word TOUGH in the dictionary.

50. The streets of a city are arranged like the lines of a chessboard. There are $$m$$ streets running north and south and $$n$$ East and West. Find the number of ways in which a man can travel from the N.W. to S.E. corner, going the shortest possible distance.