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The first body increases its rate of traveling 2 miles every hour; and the second doubles its rate every hour; then

2+2(12—1)=24, is the number of miles the first went in the last hour (176); and

(2+24)×12=156, is the whole number of miles the first went,

(180). Also,

2x211-4096, is the number of miles the second went in the last hour (186); and (4096×2-2)

(2-1)=8196, is the whole number of miles the

second went.

Therefore, 156+8190=8346 miles, the distance between them.

33. A and B set out at the same time to meet each other. A travels 3, 4, 5, &c., miles on successive days, and B 3, 4, 63, &c., miles on successive days. They meet in 10 days. What is the distance between the two places from which they traveled?

In the first series of numbers the common difference is 1; and in 41 the second series, the ratio is 43 = 13

3

A traveled on the last day

3+1(10−1)=12 miles, (176).

(3+12)

A traveled altogether

2

B traveled on the last day

3X(11)=

B traveled altogether

59049

512

×10=75 miles, (180).

59049

512

miles, (186).

×1} −3)÷(1}−1)=339 §¦7 miles, (190).

Therefore, 75+339-414 07 miles is the distance between the places.

PROBLEMS

In Permutations and Combinations.

1. In how many different ways might a company of 10 persons seat themselves around a table?

Substituting 10 for n in the Formula (194), we have

10 × 9×8×7×6×5×4×3×2×1=3628800 ways.

2. How many different numbers might be expressed by the 10 numeral figures, if 5 figures be used in each number?

10×9x8x7×6=30240, (194).

3. In how many different ways may the names of the 12 months of the year be arranged one after another?

12 × 11 × 10 × 9×8×7×6×5×4×3×2×1=479001600, (194).

*

4. How many different permutations of 8 men could be formed out of a company consisting of 15 men?

15 × 14 × 13 x 12 x 11 x 10 x 9x8=259459200, (194).

5. In how many different ways might the seven prismatic colors, red, orange, yellow, green, blue, indigo, and violet, have been arranged in the solar spectrum?

7×6×5×4×3×2×1=5040, (194).

6. How many different combinations of two colors could be formed out of the seven prismatic colors?

=21, (196).

7. How many different combinations of 5 letters may be formed out of the 26 letters of the Alphabet?

26 × 25 × 24 × 23 × 22
1×2×3×4×5

7X6

1X2

=65780, (196).

8. How many different combinations of 2 elements might be formed out of the 56 elements described in Chemistry?

56 × 55

1x 2

=1540, (196).

9. In how many different ways might a company of 20 men be arranged, in single file, in a procession?

20 × 19 × 18 x 17 x 16 x 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7×6×5 ×4×3×2×1=2432902008176640000,

(194).

10. A farmer wishes to select a team of 6 horses out of a drove containing 10 horses. How many different choices for the team will he be able to make?

=210, (196).

11. In how many different ways might the planets Mercury, Venus, the Earth, Mars, Jupiter, Saturn, Uranus, and Neptune, succeed one another in the solar system?

10 × 9 × 8 × 7×6×5
1 X2 X3 X4 X5 X6

8X7X6X5X4x3x2x1=40320, (194).

12. A company of 20 persons engaged to remain together so long as they might be able to combine in different couples in their evening walks. What time will be required to fulfil the engagement?

190 days, (196).

20 × 19

1 x 2

13. How many different permutations of 7 letters might be formed out of the 26 letters of the Alphabet?

26 x 25 x 24 x 23 x 22 x 21 x 20=3315312000, (194).

14. In an exhibition of a Public School, 5 speakers are to be taken from a class of 15 students. How many different selections of the 5 might be made? and in how many different ways might the 5 succeed one another in the delivery of their speeches?

15 × 14 × 13 × 12 × 11
1×2×3×4×5

=3003 selections, (196).

5x4x3x2x1 =120 ways, (194).

15. Out of a company consisting of 100 soldiers, six are to be taken for a particular service. How many different selections of the 6 might be made? and in how many different ways might the 6 chosen be dis posed with regard to the order of succession?

100 × 99 × 98 × 97 × 96 × 95
1 x 2×3×4 × 5 × 6

=1192052400 selections, (196)

6×5×4×3×2×1=720 ways, (194).

EXERCISES

On Rationalization and Pure Equations.

1. Find the value of x in the Equation

24-√/2x2+9=15.

By transposition, −√2x2+9=15—24.
Adding similar terms and squaring both sides,
2x2+9=81;

from which we shall find x= ±6, (256).

2. Find the value of x in the Equation
13-√/3x2+16 5.

=

3x2+16=5—13.

By transposition,
Adding similar terms and squaring both sides,

3x2+16=64;

from which we shall find x=±4, (256).

3. Find the value of x in the Equation
35+x-5=40.

[ocr errors]

By transposition, 3x-5=40-35.
Adding similar terms, and cubing both sides
x-5=125;

which will give x= =130.

4. Find the value of x in the Equation
1+2√x =√4x+21.

=

By transposition, 2x4x+21 −1.
Squaring both sides,

4x=4x+21-2√√/4x+21 +1.

Transposing, and adding similar terms,
2v4+21 =22.

Squaring both sides

16x+84 484.

From this Equation we shall find x=25.

5. Find the value of x in the Equation

√x−32=√/x-√32.

Squaring both sides,

x-32=x-√32x +8.

Transposing, and adding similar terms,

√32x=40.

Squaring both sides,

This Equation will give x=50.

6. Find the value of x in the Equation

[ocr errors]

32x=1600.

3+√x+4 × √x-4=10.

Multiplying together the two radical quantities in this Equation,

3+√x2-16=10.

Transposing, and uniting similar terms

x2-16=7.

x2-16=49;

Squaring both sides,

which will give x=±√/65.

7. Find the value of x in the Equation

a+√x−3 × √x+3=4a.

Multiplying together the two radical quantities in this Equation,

a+√x2-9=4a.

Transposing, and adding similar terms,

√x2-9=3a.

Squaring both sides,

By transposition,

Hence, we have =√√9a2+9=±3√/a2+1, (237).

8. Find the value of x in the Equation

x2-9=9a2.

x2-9a2+9.

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