# 64. Permutations and Combinations Problems Part 6¶

1. For all positive integers show that $$2.6.10.\ldots.(4n-6) = (n + 1) (n + 2)\ldots(2n - 1)2n$$

2. Show that $$^{47}C_4 + \sum_{i=0}^350-{}^iC_3 + \sum_{j=1}^5{}^{56 - j}C_{53} = {}^57C_4$$

3. Show that $$^nC_k + \sum_{j=0}^m{}^{n+j}C_k - 1 ={}^{n+m - 1}C_k$$

4. Show that $${}^mC_1 +{}^mC_2 + \ldots +{}^{m + n - 1}C_n = ^nC_1 +{}^{n + 1}C_2 + \ldots +{}^{n + m - 1}C_m$$

5. How many numbers divisible by $$25$$ can be made with the digits $$0, 1, 2, 3, 4, 5, 6$$ and $$7$$?

6. How many numbers divisible by $$4$$ can be made with the digits $$1, 2, 3, 4$$ and $$5$$?

7. How many numbers of $$4$$ digits can be made with the digits $$0, 1, 2, 3, 4, 5$$ which are divisible by $$3,$$ digits being unrepeated in the same number? How many of these will be divisible by $$6$$?

8. Find the sum of all the $$4$$ digit numbers formed with the digits $$1, 3, 3, 0$$?

9. Show that the number of permutation of $$n$$ different things taken not more than $$r$$ at a time, when each thing may be repeated any number of times is $$\frac{n(n^r - 1)}{n - 1}$$

10. How many different $$7$$ digit numbers are there sum of whose digits is even?

11. $$k$$ numbers are chosen with replacement from the numbers $$1, 2, 3, \ldots, n$$. Find the number of ways of choosing the numbers so that the maximum number chosen chosen is exactly $$r(r\leq n)$$.

12. Find the number of $$n$$ digit numbers formed with the digits $$1, 2, 3,\ldots, 9$$ in which no two consecutive digits repeat.

13. A valid FORTRAN identifier consists of a string of one to six alphanumeric characters which are $$A, B,\ldots,Z$$ $$0, 1, 2,\ldots,9$$ beginning with a letter. How many valid FORTRAN identifiers are there?

14. Find the number of five digit numbers which can be made with at least one repeated digit.

15. Find the number of numbers between $$2\times 10^4$$ and $$6\times 10^4$$ having sum of digits even.

16. Find the number of ways in which the candidates $$A_1, A_2,\ldots, A_{10}$$ can be ranked,

1. if $$A_1$$ and $$A_2$$ are next to each other.

2. if $$A_1$$ is always above $$A_2$$

17. $$n + m$$ chairs are placed in a line. You have to seat $$n$$ men and $$m$$ women on these chairs such that no man gets a seat between two women. In how mnay ways can these people be seated?

18. How many words can be made with the letters of the word INTERMEDIATE if no vowel is between two consonants?

19. In how many ways can $$5$$ identical black balls, $$7$$ identical red balls and $$6$$ identical green balls be arranged so that at least one ball is separated from balls of the same color?

20. Ten guests are to be seated in a row of which three are ladies. The ladies insist on sitting together while tow of the gentlemen refuse to take consecutive seats. In how many ways can the guests be seated?

21. Show that number of permutations of $$n$$ different things taken all at a time in which $$p$$ particular things are never together is $$n! - (n - p + 1)!p!$$

22. Find the number of ways in which six ‘+’ signs and four ‘-‘ signs can be arranged in a line so that no two ‘-‘ signs occur together.

23. In how many ways can $$3$$ ladies and $$5$$ gentlemen arrange themselves about a round so that every gentleman may have a one lady by his side.

24. How many words of seven letters can be formed by using the letters of the word SUCCESS so that

1. the two Cs are together but not the two S.

2. nether the two C nor the two S are together.

25. A dictionary is made of the words that can be formed from the letters of the word MOTHER. What is the position of the word MOTHER in that dictionary if words a printed in the same order as than of an ordinary dictionary?

26. A train going from Kolkata to Delhi stops at $$7$$ intermediate stations. Five persons enter the train during the journey with five different tickets of the same class. How many different set of tickets they could have had?

27. A train going from Cambridge to London stops at nine intermediate stations. Six persons enter the train during the journey with six different tickets of the same class. How many different set of tickets they could have had?

28. In how many ways can clear and cloudy days occur in a week? It is given that any day is entirely either clear or cloudy.

29. A student is allowed to select at most $$n$$ books from a collection of $$2n + 1$$ books. If the total no. of ways in which he can select at least one book is $$63$$, find the value of $$n$$.

30. There are $$m$$ bags which are numbered by $$m$$ consecutive integers starting with the number $$k.$$ Each bag contains as many different flowers as the number marked on the bag. A boy has to pick up $$k$$ flowers from any one of the bags. In how many different ways can he do it?

31. How many committees of $$11$$ persons can be made out of $$50$$ persons if three particular persons are not to be included together?

32. There are $$m$$ intermediate stations on a railway line between two places P and Q. In how many ways can the train stop at three of these intermediate stations no two of which are consecutive?

33. A is a set containing $$n$$ elements. A subset P of A is chosen. The set A is reconstructed by replacing the elements of P. Then a subset Q of A is chosen. Find the number of ways of choosing P and Q such that

1. $$P\cap Q$$ contains exactly $$2$$ elements

2. $$P\cap Q=\phi$$

34. $$A$$ is a set containing $$n$$ elements. A subset $$P_1$$ is chosen. The set A is reconstructed by replacing the elements of $$P_1.$$ Then a subset $$P_2$$ is chosen and again the set is reconstructed by replacing elements of $$P_2.$$ In this way $$m$$ subsets $$P_1, P_2,\ldots,P_m$$ are chosen, where $$m > 1.$$ Find the number of ways of choosing these subsets such that

1. $$P_1\cup P-2\cup \ldots \cup=P_m$$ contains all the elements of $$A$$ except one.

2. $$P_1\cup P-2\cup \ldots \cup=P_m = A$$

3. $$P_1\cap P-2\cap \ldots \cap=P_m = \phi$$

35. There are three sections in a question paper, each containing $$5$$ questions. A candidate has to solve any $$5$$ questions, choosing at best one from each section. Find the number of ways in which the candidate can choose the questions.

36. Two numbers are selected at random from $$1, 2, 3, \ldots, 100$$ and are multiplied. Find the number of ways in which the two numbers can be selected so that the product thus obtained is divisible by $$3.$$

37. In how many ways can a mixed doubles game in tennis be arranged from $$5$$ married couples, if no husband and wife play in the same game.

38. There are $$n$$ concurrent lines and another line parallel to one of them. How many different triangles will be formed by the $$(n + 1)$$ lines?

39. In a plane there are $$n$$ lines no two of which are parallel and no three are concurrent. How many different triangles can be formed with their points of intersection as vertices?

40. The England cricket eleven is to be selected out of fifteen players, five of them are bowlers. In how many ways can the team be selected so that the team contains at least three bowlers?

41. There are two bags each containing $$m$$ balls. Find the number of ways in which equal no. of balls can be selected from both bags if at least one ball from each bag is to be selected.

42. A committee of $$12$$ is to be formed from $$9$$ women and $$8$$ men. In how many ways can this be done if at least $$5$$ women have to be included in a committee? In how many of these committees

1. the women are in majority

2. the men are in majority

43. $$m$$ equi-spaced horizontal lines are intersected by $$n$$ equi-spaced vertical lines. If $$m<n$$ and the distance between two successive horizontal lines is same as the distance between two successive vertical lines, show that the number of squares formed by these lines is $$\frac{1}{6}m(m - 1)(3n - m - 1)$$

44. There are two sets of parallel lines, their equations being $$x\cos \alpha + y\sin\alpha = p; p = 1, 2, 3, \ldots, m$$ and $$y\cos\alpha - x\sin\alpha=q; q = 1, 2, 3, \ldots, n(n > m)$$ where $$\alpha$$ is a constant. Show that the lines form $$\frac{1}{6}m(m - 1)(3n - m - 1)$$ squares.

45. In how many different ways can a set $$A$$ of $$3n$$ elements be partitioned in $$3$$ subsets of equal number of elements?

46. In how many ways $$50$$ different things can be divided in $$5$$ sets three of them having $$12$$ things and two of them having $$7$$ things each?

47. In how many ways $$50$$ different things can be distributed among $$5$$ persons so that, three of them get $$12$$ things each and two of them get $$7$$ things each?

48. If $$a, b, c, \ldots, k$$ are positive integers such that $$a + b + c + \ldots + k \leq n,$$ show that $$\frac{n!}{a!b!\ldots k!}$$ is a positive integer.

49. If $$a, b \in N,$$ show that $$\frac{(ab)!}{a!(b!)^a}$$ is an integer.

50. If $$n\in N,$$ sho that $$\frac{(n^2)!}{(n!)^{n + 1}}$$ is an integer.