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Ex. 3. Shew that n straight lines, no two of which are parallel and no three of which meet in a point, divide a plane into n (n + 1) +1 parts.

The nth straight line is cut by each of the other n1 lines; and hence it is divided into n portions. Now there are two parts of the plane on the two sides of each of these portions of the nth line which would become one part if the nth line were away. Hence the plane is divided by n lines into n more parts than it is divided by n - 1 lines.

Hence, if F(x) be put for the number of parts into which the plane is divided by x straight lines, we have

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Ex. 4. Suppose n things to be given in a certain order of succession. Shew that the number of ways of taking a set of three things out of these, with the condition that no set shall contain any two things which were originally contiguous to each other is † (n − 2) (n − 3) (n − 4). Shew also that if the n given things are arranged cyclically, so that the nth is taken to be contiguous to the first, the number of sets is reduced to in (n −4) (n − 5).

Consider the second case first.

Let the different things be represented by the letters a, b, c,...... k, l.

Suppose that a is taken first. Then, if either of the two letters next but one to a be taken second, any one of n-5 letters can be taken for the third of the set. If, however, the second letter is not next but one to a, but in either of the n- 5 other possible places, there would be a choice of n-6 places for the third letter of the set. Hence the total number of ways of taking 3 letters in order a being first is 2 (n-5) + (n − 5) (n − 6), that is (n-4) (n-5). There is the same number when any one of the other letters is taken first; hence, as the order in which the three letters in a set are taken is indifferent, the total number of sets is n (n − 4) (n − 5).

In order to obtain the first case from the second, we have only to suppose that a and I are no longer contiguous. Hence the number in the first case is the same as that in the second with the addition of those sets which contain a and 7, and there are n-4 of these. Hence the number in the first case is n (n − 4) (n − 5) + (n − 4) = (n-2) (n-3) (n-4).

Ex. 5. There are n letters and n directed envelopes: in how many ways could all the letters be put into the wrong envelopes?

Let the letters be denoted by the letters abc... and the corresponding envelopes by a', b', c',......

Let F (n) be the required number of ways.

Then a can be put into any one of the n 1 envelopes b', c',.... Suppose a is put into k'; then k may be put into a', in which case there will be F (n-2) ways of putting all the others wrong. Also if a is put into k', the number of ways of disposing of the letters so that k is not put in a', b not in b', &c. is F (n-1).

Hence the number of ways of satisfying the conditions when a is put into k' is F(n − 1) + F (n-2). The same is true when a is put into any other of the envelopes b', c',... Hence we have

F(n)=(n-1) {F (n − 1) + F (n − 2) };

F(n)-n F (n-1)= {F (n − 1) − (n - 1) F (n-2)}.

Similarly F(n-1) (n - 1) F (n-2)= {F (n-2)-(n-2) F (n-3)}

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1. In how many different ways may twenty different things be divided among five persons so that each may have four?

2. A crew of an eight-oar has to be chosen out of eleven men, five of whom can row on the stroke side only, four on the bow-side only, and the remaining two on either side. How many different selections can be made?

3. There are three candidates for a certain office and twelve electors. In how many different ways is it possible for them all to vote; and in how many of these ways will the votes be equally divided between the candidates?

4. Shew that CC is equal to

4n 2n

2n 12

1.3.5......(4n-1)

{1.3.5...... (2n − 1)}**

5. Find the number of significant numbers which can be formed by using any number of the digits 0, 1, 2, 3, 4, but using each not more than once in each number.

6. Shew that in the permutations of n things r together, the number of permutations in which p particular things occur is -P-pPp

7.

There are n points in a plane, no three of which are in the same straight line; find the number of straight lines formed by joining them.

8. There are n points in a plane, of which no three are on a straight line except m which are all on the same straight line. Find the number of straight lines formed by joining the points.

9. There are n points in a plane, of which no three are on a straight line except m which are all on a straight line. Find the number of triangles formed by joining the points.

10. Shew that the number of different n sided polygons formed by n straight lines in a plane, no three of which meet in a point, is n − 1.

11. There are n points in a plane which are joined in all possible ways by indefinite straight lines, and no two of these joining lines are parallel and no three of them meet in a point. Find the number of points of intersection, exclusive of the n given points.

12. Through each of the angular points of a triangle m straight lines are drawn, and no two of the 3m lines are parallel. Find the number of points of intersection.

13. The streets of a city are arranged like the lines of a chess-board. 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. corner to the S.E. corner, going the shortest possible distance.

14. How many triangles are there whose angular points are at the angular points of a given polygon of n sides but none of whose sides are sides of the polygon?

15. Shew that 2n persons may be seated at two round tables, n persons being seated at each, in

12n

different ways.

n

2

16. A parallelogram is cut by two sets of m lines parallel to its sides shew that the number of parallelograms thus formed is (m + 1)2 (m + 2)3.

17. Find the number of ways in which p positive signs and n negative signs may be placed in a row so that no two negative signs shall be together.

18. Shew that the number of ways of putting m things in n+ 1 places, there being no restriction as to the number in each place is (m + n)! / m ! n!

19. Shew that 2n things can be divided into groups of n pairs in

12n

2" n

ways.

20. Find the number of ways in which mn things can be divided into m sets each of n things.

21.

Shew that n planes through the centre of a sphere, no three of which pass through the same diameter, will divide the surface of the sphere into n2 - n + 2 parts.

S. A.

19

22. Shew that the number of parts into which an infinite plane is divided by mn straight lines, m of which pass through one point and the remaining n through another, is mn + 2m + 2n − 1, provided no two of the lines be parallel or coincident.

23. Find the number of parts into which a sphere is divided by m + n planes through its centre, m of which pass through one diameter and the remaining n through another, no plane passing through both these diameters.

24. Find the number of parts into which a sphere is divided by a+b+c+... planes through the centre, a of the planes passing through one given diameter, b through a second, c through a third, and so on; and no plane passing through more than one of these given diameters.

25. Show that n planes, no four of which meet in a point, divide infinite space into † (n3 + 5n + 6) different regions.

26. Prove that if each of m points in one straight line be joined to each of n points in another, by straight lines terminated by the points; then, excluding the given points, the lines will intersect mn (m − 1) (n − 1) times.

27. No four of n points lying in a plane are on the same circle. Through every three of the points a circle is drawn, and no three of the circles have a common point other than one of the original n points. Shew that the circles intersect in 12n (n - 1) (n − 2) (n − 3) (n − 4) (2n-1) points besides the original n points, assuming that every circle intersects every other circle in two points.

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