# 69. Permutations and Combinations Problems Part 5

1. There are $$n$$ letters and $$n$$ corresponding envelops. In how many ways the letter be placed in envelopes(one letter in each envelop) so that no letter is put in right envelope.

2. Find the number of non-congruent rectangles that can be formed on a chessboard.

3. Show that the no. of ways in which three numbers in A. P. can be selected from $$1, 2, 3, \ldots , n$$ is $$\frac{1}{4}(n-1)^2$$ or $$\frac{1}{4}n(n - 2)$$ according as $$n$$ is odd or even.

4. two packs of $$52$$ playing cards are shuffled together. Find the number of ways in which a man can be dealt $$26$$ cards so that he does not get two cards from same suit and same denomination.

5. There is a polygon of $$n$$ sides($$n>5$$). Triangles are formed by joining the vertices of the polygon. How many triangles are there? Also, prove that number of these triangles which have no side common with any of the sides of the polygon is $$\frac{1}{6}n(n-4)(n-5)$$.

6. $$n$$ different objects are arranged in a row. In how many ways can $$3$$ objects be selected so that

1. all the three objects are consecutive

2. all the three objects are not consecutive

3. no two objects are consecutive

7. There are $$12$$ intermediate stations between two places $$A$$ and $$B$$. In how many ways can a train be made to stop at $$4$$ of these $$12$$ intermediate stations no two of which are consecutive?

8. There are $$m$$ points in a plane which are joined by straight lines in all possible ways and of these no two are coincident or parallel and no three of them are concurrent except at the points. Show that the number of points of intersection, other than the given points of the lines so formed is $$\frac{m!}{8.(m-4)!}$$

9. Find the number of ways of choosing $$m$$ coupons out of an unlimited number of coupons bearing the letters A, B and C so that they cannot be used to spell the word BAC.

10. A straight is a five-card hand containing consecutive values. How many different straights are there? If the cards are not all from the same suit, then how many straights are there?

11. $$A$$ is a set containing $$n$$ elements. A subset $$P_1$$ of $$A$$ is chosen. The set $$A$$ is reconstructed by replacing the elements of $$P_1$$. Then a subset $$P_2$$ of $$A$$ is chosen and again set $$A$$ is reconstructed by replacing the elements of $$P_2$$. In this ways $$m$$ subsets are chosen, where $$m>1$$

Find the no. of ways of choosing $$P_1, P_2, \ldots, P_m$$ such that

1. $$P_1\cup P_2\cup \ldots\cup P_m$$ contains exactly $$r$$ elements of $$A$$

2. $$P_1\cap P_2\cap \ldots\cap P_m$$ contains exactly $$r$$ elements of $$A$$

3. $$P_i\cap P_j = \phi$$ for $$i\neq j$$

12. Find the number of ways in which $$m$$ identical balls be distributed among $$2m$$ boxes so that no box contains more than one ball and show that it lies between $$\frac{4^m}{\sqrt{2m + 1}}$$ and $$\frac{4^m}{2\sqrt{m}}$$

13. From $$6$$ gentlemen and $$4$$ ladies, a committee of $$5$$ is to be formed. In how many ways can this be done if the committee is to include at least one lady and if two particular ladies refuse to serve on the same committee?

14. A man has $$7$$ relatives, $$4$$ of them are ladies and $$3$$ are gentlemen. His wife also has $$7$$ relatives, $$3$$ of them are ladies and $$4$$ are gentlemen. In how many ways can they invite a dinner party of $$3$$ ladies and $$3$$ men so that there are $$3$$ of the man’s relatives and $$3$$ of the wife’s relatives?

15. Prove that if each of $$m$$ points on one straight line be joined to each of $$n$$ points on the other straight line terminated by the points, then excluding the points given on the two lines, number of points of intersection of these lines is $$\frac{1}{4}mn(m-1)(n-1)$$.

16. John has $$x$$ children by his first wife. Mary has $$x + 1$$ children by her first husband. They marry and have children of their own. The whole family has $$24$$ children. Assuming that two children of same parents do not fight prove that maximum possible no. of ways fight can take place is $$191$$.

17. Find the number of divisors and sum of the divisors of $$2520$$

18. Five balls of different colors are to be places in three boxes of different sizes. Each box can hold all five balls. In how many different ways can we place the balls so that no box remains empty.

19. Prove that $$(n!)!$$ is divisible by $$(n!)^{(n- 1)!}$$

20. If $$a$$ and $$b$$ are positive integers, show that $$\frac{(ab)!}{a!(b!)^a}$$ is an integer.

21. A conference attended by $$200$$ delegates is held in a hall. The hall has seven doors, marked $$A, B, C, \ldots, G$$. At each door, an entry book is kept and the delegates entering that door sign in it in the order in which they enter. If each delegate is free to enter any time and through any door he likes, how many different sets of seven lists would arise in all?

22. In how many ways $$16$$ identical things can be distributed among $$4$$ persons if each person gets at least $$3$$ things?

23. Show that a selection of $$10$$ balls can be made from an unlimited number of red, white, blue and green balls in $$286$$ different ways and that $$84$$ of these contain balls of all four colors.

24. In how many ways $$30$$ marks can be allotted to $$8$$ questions if each question carries at least $$2$$ marks?

25. In an examination the maximum marks for each of the three papers are $$50$$ each. Maximum marks for the fourth paper is $$100$$. Find the number of ways in which a student can score $$60%$$ marks in the aggregate.

26. Let $$n$$ and $$k$$ be positive integers such that $$n\geq \frac{k(k+1)}{2}$$. Find the number of solutions $$(x_1, x_2, \ldots, x_k), x_1\geq1, x_2\geq 2, \ldots, x_k\geq k$$ all satisfying $$x_1 + x_2 + \ldots + x_k=n$$

27. Find the number of integral solutions of equation $$x + y + z + t = 29, x>0, y>1, z>2$$ and $$t\geq 0$$

28. Find the number of non-negative integral solutions of $$x + y + z + 4t = 20$$.

29. Find the number of non-negative integral solutions to the system of equations $$x + y + z + u + t = 20$$ and $$x + y + z = 5$$.

30. Find the number of positive integral solutions of the inequality $$3x + y + z \leq 30$$.

31. Find the number of positive unequal integral solution of the equation $$a + b + c + d = 20$$.

32. How many integers between $$1$$ and $$1,000,000$$ have the sum of digits $$18$$?

33. Prove that the number of combinations of $$n$$ letters together out of $$3n$$ letters of which $$n$$ are $$a$$ and $$n$$ are $$b$$ and the rest unlike is $$(n + 2)2^{n - 1}$$

34. An eight-oared boar is to be manned by a crew chosen from $$11$$ men of whom $$3$$ can steer but cannot row and the rest cannot steer. In how many ways can the crew be arranged if two of them can only row on bow side?

35. Find the total number of ways of selecting five letters from the letters of the word “INDEPENDENCE”.

36. Find the number of combinations and the number of permutations of the letters of the word “parallel”, taken four at a time.

37. Find the value of $$n$$ for which $$\frac{^{n+4}P_4}{(n+2)!} - \frac{143}{4.n!} < 0$$

38. Find the values of $$n$$ for which $$\frac{195}{4.n!} - \frac{(n+3)(n+2)(n+1)}{(n+1)!} > 0$$

39. If $${}^{n-2}P_4:{}^{n+2}C_8 = \frac{16}{57}$$, find the value of $$n$$

40. If $$^nP_r ={}^nP_{r+1}$$ and $$^nC_r={}^nC_{r - 1}$$, find $$n$$ and $$r$$.

41. If $$^nP_{r-1}:{}^nP_r:{}^nP_{r+1} = a:b:c$$, prove that $$b^2 = a(b + c)$$

42. If $$^{n+1}C_{r+1}:{}^nC_r:{}^{n-1}C_{r-1} = 11:6:3$$, find the values of $$n$$ and $$r$$.

43. Show that $$\sum_{k=m}^n{}^kC_r ={}^{n+1}C_{r+1} -{}^mC_{r+1}$$

44. Show that $$^nC_{n-r} + 3.{}^nC_{n-r} +{}^nC_{n-r+4} +{}^nC_{n-r+3} = {}^{n+3}C_r$$

45. Find $$r$$ for which $${}^{18}C_{r-2} + 2{}^{18}C_{r-1} + {}^{18C_r}\geq{}^{20}C_{13}$$

46. Prove that $${4n\choose 2n}:{2n\choose n} = {1.3.5\ldots(4n-1)}: {1.3.5.\ldots(2n-1)}^2$$

47. Find the positive integral values of $$x$$ such that $$^{x-1}C_4 -{}^{x-1}C_3 - \frac{5}{4}(x-2)(x-3)<0$$

48. Prove that $$^{2n}P_n = 2^n{1.3.5\ldots(2n-1)}$$

49. Show that there cannot exist two positive integers $$n$$ and $$r$$ for which $$^nC_r,{}^nC_{r+1},{}^nC_{r+2}$$ are in G.P.

50. Show that two positive integers $$n$$ and $$r$$ cannot be found such that $$^nC_r,{}^nC_{r+1},{}^nC_{r+2},{}^nC_{r+3}$$ are in A.P.