21. The sum of two whole numbers is 100; find the chance that their product is greater than 1000.

22. The sum of two positive quantities is given; prove that it is an even chance that their product will not be less than three-fourths of their greatest product; prove also that the chance of their product being less than one-half their

23. Two men A and B have a and b counters respectively, and they play a match consisting of separate games, none of which can be drawn, and the winner of a game receives a counter from the loser. The two players have an equal chance of winning any single game, and the match is continued until one of the players has no more counters. Shew that A's chance of winning the match is

a

a+b

24. An urn contains a number of balls which are known to be either white or black, and all numbers are equally likely. If the result of p + q drawings (the balls not being replaced) is to give p white and 9 black balls, shew that the chance that the

next drawing will give a black ball is

9+1

p+q+2°

25. Two sides play at a game in which the total number of points that can be scored is 2m + 1; and the chances of any point being scored by one side or the other are as 2m+ 1 x to 2m + 1−y, where x and y are the points already scored by the respective sides. Shew that the chance that the side which scores the first point will just win the game is

(2m! 2m+1!)2

(m!)3 m + 1! 4m + 1 ! '

CHAPTER XXXI.

DETERMINANTS.

409. IF there are nine quantities arranged in a square as under:

then all the possible products of the quantities three together, subject to the condition that of the three quantities in each product one and only one is taken from each of the rows and one and only one from each of the columns, will be

Let now these products be considered to be positive or negative according as there is an even or an odd number of inversions of the natural order in the suffixes; then the algebraic sum of all the products will be

a,b,c, — a,b,c, + a,b,c, — a,b,c, + a,b,c, — a,b,c, ....(A);

for there are no inversions in a,b,c,, there is one inversion in a,b,c, since 3 precedes 2, there are two inversions in a,b,c, since 2 and 3 both precede 1, there is one inversion in a,b,c, since 2 precedes 1, there are two inversions in a,b,c, since 3 precedes both 1 and 2, and there are three inversions in a,b,c, since 3 precedes both 1 and 2 and 2 precedes 1.

The expression (A) is called the determinant of the nine quantities a, a,, &c., which are called its elements; and the products a,b,c,, a,b,c,, &c., are called the terms of the determinant.

the members of the same row being distinguished by the same letter, and the members of the same column by the same suffix; and if all the possible products of the quantities n at a time are taken subject to the condition that of the n quantities in each product one and only one is taken from every row and one and only one from every column, and if the sign of each product is considered to be positive or negative according as there is an even or an odd number of inversions of the natural order in the suffixes; then the algebraic sum of all the products so formed is called the determinant of the n2 quantities or elements.

To denote that the n2 quantities are to be operated upon in the manner above described, they are enclosed by two lines, as in the above scheme.

The diagonal through the left-hand top corner is called the principal diagonal; and the product of the n elements a1, b2, C3, . . . . . ., m, which lie along it, is called the principal term of the determinant.

2'

n

All the other terms can be formed in order from the principal term by taking the letters in their alphabetical order and permuting the suffixes in every possible way: on this account a determinant is sometimes represented by enclosing its principal term in brackets; thus the above determinant would be written [a,b,c,...m,], the

determinant is also often represented by the notation Σ (± a,b,cg...mn).

When only one determinant is considered it is generally denoted by the symbol A.

A determinant is said to be of the nth order when there are n elements in each of its rows or columns, and therefore also n elements in each of its terms.

411. Since there are as many terms in a determinant of the nth order as there are permutations of the n suffixes, it follows that there are n terms in a determinant of the nth order. There are, for example, six terms in a determinant of the third order.

412. The law by which the sign of any term of a determinant is found is equivalent to the following:

Take the elements in order from the successive rows beginning at the first; then the sign of any term is positive or negative according as there is an even or an odd number of inversions in the order of the columns from which the elements are taken.

2

We will now shew that the words row and column may be interchanged in the above law. To prove this, consider any product, for example, a,b,c,def and its equivalent cfb,dae, where in the first form the letters follow the alphabetical order and in the second form the numbers follow the natural order.

We have to shew that the number of inversions in the suffixes in the first form is the same as the number of inversions of the alphabetical order in the second form. This follows immediately from the fact that if, in the first form, any suffix follow r suffixes greater than itself; then, in the second form, the letter coresponding to that suffix must preceder letters earlier than itself in alphabetical order. Thus, in the example, 2 follows four suffixes greater than itself in a,b,c,deƒ, and ƒ precedes four letters earlier than itself in cfbda e

3 1 4 6

Since the words rows and columns are interchangeable in the law which determines the sign of any term, we have the following

Theorem. A determinant is unaltered by changing its rows into columns and its columns into rows.

Ex. 1. Count the number of inversions in 2314, 3142 and 4231.

Ans. 2, 3, 5.

Ex. 2. Count the number of inversions in 4132, 35142 and 531264.

Ans. 4, 6, 7.

Ex. 3. What are the signs of the terms bfg, cdh and ceg in the determinant a b C?

Ex. 4. What are the signs of the terms bgiq, celn and dfkm in the determinant a b с d?

e f g h

i j k

m n p q

[The order of the columns is 2314, 3142 and 4231.]

Ans. +,

413. Theorem I. If in any term of a determinant any two suffixes be interchanged, another term of the determinant will be obtained whose sign is opposite to that of the original

term.

Let P.ha.kg be any term of a determinant, P being the product of all the elements except ha and kg; then, by interchanging a and B we have P.he.ka. Now since P.ha.ke is a term of the determinant, P can contain no element from the rows of h's and k's and no element from

α