Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

2. How many different numbers can be made of the following figures, 1220005555 ? Ans. 12600.

3. What is the variety in the succession of the following musical notes, fa, fa, fa, sol, sol, la, mi, fa? Ans. 3360.

PROBLEM IV.

To find the changes of any given number of things, taken a given number at a time; in which there are several given things of one sort, several of another, &c..

RULE.*

1. Find all the different forms of combination of all the given things, taken as many at a time as in the question.

2. Find the number of changes in any form, and multiply it by the number of combinations in that form.

3. Do the same for every distinct form; and the sum of all the products will give the whole number of changes required.

NOTE. To find the different forms of combination proceed thus :

1. Place the things so, that the greatest first, and the rest in order.

indices may

be

2. Begin with the first letter, and join it to the second, third, fourth, &c. to the last.

3. Then take the second letter, and join it to the third, fourth, &c. to the last; and so on through the whole, always remembering to reject such combinations as have occurred before; and this will give the combinations of all the twos.

4. Join

* The reason of this rule is plain from what has been shewn before, and the nature of the problem.

4. Join the first letter to every one of the twos following it, and the second, third, &c. as before; and it will give the combinations of all the threes.

5. Proceed in the same manner to get the combinations of all the fours, &c. and you will at last get all the several forms of combination, and the number in each form.

EXAMPLES.

1. How many changes may be made of every 4 letters, that can be taken out of these 6, aaabbc ?

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small]

38 the number of changes required.

2. How many changes can be made of every 8 letters out of these 10, aaaabbccde?

Ans. 22260.

3. How many different numbers can be made out of I unit, 2 twos, 3 threes, 4 fours, and 5 fives, taken 5 at a 3 time?

Ans. 2111

PROBLEM

PROBLEM V.

To find the number of combinations of any given number of things, all different from one another, taken any given number at a time.

[ocr errors]

RULE.

1. Take the series 1, 2, 3, 4, &c. up to the number to be taken at a time, and find the product of all the terms.

[blocks in formation]

m-3 4

->

&c. to n terms; where m is the number of given quanti

ties, and n those to be taken at a time.

DEMONSTRATION OF THE RULE. 1. Let the number of things to be taken at a time be 2, and the things to be combined =m.

Now, when m, or the number of things to be combined, is only two, as a and b, it is evident, that there can be only one combination, as ab; but if m be increased by 1, or the letters to be combined be 3, as abc, then it is plain, that the number of combinations will be increased by 2, since with each of the former letters, a and b, the new letter e may be joined. It is evident, therefore, that the whole number of combinations, in this case, will be truly expressed by 1+2.

Again, if m be increased by one letter more, or the whole number of letters be four, as abcd; then it will appear, that the whole number of combinations must be increased by 3, since with each of the preceding letters the new letter d may be combined. The combinations, therefore, in this case, will be truly expressed by 1+2+3.

In

2. Take a series of as many terms, decreasing by 1, from the given number, out of which the election is to be made, and find the product of all the terms.

3. Divide the last product by the former, and the quo tient will be the number sought.

A-XI EXAMPLES.

In the same manner, it may be shewn, that the whole number of combinations of 2, in 5 things, will be 1+2+3+4; of 2, in 6 things, 1+2+3+4+5; and of 2, in 7, 1+2+3+4+5 +6, &c.

Whence, universally, the number of combinations of m things, táken 2 by 2, is 1+2+3+4+5+6, &c. to my terms. But the sum of this series is. +. ; which is the same as

the rule.

m

m- I

2. Let now the number of quantities in each combination be supposed to be three.

Then it is plain, that when m≈3, or the things to be combined are abc, there can be only one combination; but if m be increased by 1, or the things to be combined be 4, as abcd, then will the number of combinations be increased by 3; since 3 is the number of combinations of 2 in all the preceding letters abc, and with each two of these the new letter d may be combined.

The number of combinations, therefore, in this case, is 1+3. Again, if m be increased by one more, or the number of letters be supposed 5; then the former number of combinations will be increased by 6; that is, by all the combinations of 2 in the 4 preceding letters, abcd; since, as before, with each two of these the new lettere may be combined.

The number of combinations, therefore, in this case, is I +3+6.

Whence, universally, the number of combinations of m things, taken 3 by 3, is 1+3+6+10, &c. to m-z terms.

[ocr errors][merged small]

EXAMPLES.A

1. How many combinations can be made of 6 letters out of 10?

` 1×2×3×4×5 × 6(= the number to be taken at a time)=720 10X9X8X7×6×5(= same number from 10)=151200 720)151200(210 the answer.

1440

720

720

2. How many combinations can be made of 2 letters out of 24 letters of the alphabet ?

Ans. 276.

3. A general, who had often been successful in war, was asked by his King, what reward he should confer upon him for his services; the general only desired a farthing for every file, of 10 men in a file, which he could make with a body of 100 men what is the amount in pounds sterling?

Ans. 18031572350l. 98. 2d.

[merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]

And the same thing will hold, let the number of things, to be taken at a time, be what it may; therefore the number of com

[blocks in formation]
« ΠροηγούμενηΣυνέχεια »