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Find, by Case II., Art. 89, the number of combinations, without repetitions, of a number of individual things, which is one less than the number of letters, the class of the combination be. ing represented by half the number of letters after one is sub. tracted: then multiply this result by 4.
Examples. 1. How many ways may the first nine letters of the alphabet be read in this way?
Subtracting one from the number of letters, we get 9-1=8; therefore, by Case II., Art. 89, we find the number of combinations of 8 things, without repetitions, of the fourth class, to be 8X7 X 6 X5
=70; this multiplied by 4, gives 280 for the num4X3 X2X1 ber sought.
2. In how many ways. may the phrase, MODERATE YOUR CURIOSITY, be read after the above manner ?
Ans. 739024. 3. In how many ways may the 26. letters of the alphabet be read in this way?
MISCELLANEOUS QUESTIONS SOLVED BY
ANALYSIS. 92. A man and his wife usually drank out a cask of beer in 12 days; but when the man was from home, it lasted the woman 30 days. How many days would the man alone be in drisking its
Since it requires 12 days for the man and his wife to drink out the cask, they must, in each day, drink of it,
Again, since the woman is 30 days in drinking it, she must, in each day, drink zb of it..
Hence, the fractional part which the man drank in one day must be t-b=;.. in 20 days he could drink the whole.
2. A person bought several gallons of wine for $94, and after using 7 gallons himself, sold 1 of the remainder for $20. How many gallons had he at first ?
SOLUTION. Since he sold of the remainder, after using 7 gallons, for $20, he could have sold the whole of the remainder for $80: therefore, the value of the 7 gallons, which he used, wag 94-80=$14; and one gallon must have cost 4=$2. The wine being worth $2 per gallon, he must have purchased =47 gallons.
3. A person in play, lost of his money, and then won 3 shillings, after which, he lost } of what he then had; and this done, he found that he had but 12 shillings remaining. How much had he at first ?
Since, in the first place, he lost 1 of his money, he must have had left of it, to which, adding the 3 shillings which he won, he had of his money + 3 shillings. Again, he loses of this, and consequently retains , of it; so that į of 4 of his money + f of 3 shillings, or which is the same, i of his money + 2 shillings, is what he finally had remaining ; this, by the condi. tions of the question, is 12 shillings, •'. we have this relation : 1 of his money+2 shillings, must equal 12 shillings, or which is the same, i of his money must equal 10 shillings; and consequently his money must have been 20 shillings.
4. A fish was caught whose tail weighed 9 pounds, his head weighed as much as his tail and half his body, and his body weighed as much as his head and tail together. What was the weight of the fish ?
Since the head of the fish is equal to 1 of the body, together with the tail=9 pounds, it follows, that the head and tail together must equal } of the body + 18 pounds. But by the question, the head and tail together is equal to the whole body; .. we have this relation : s of the body + 18 pounds, must equal the whole body ; consequently, 1 of the body must equal 18 pounds, and the whole body is 36 pounds. And since the body weighed as much as the head and tail together, it fol. lows, that the weight of the whole fish was twice that of the body, or eight times that of the tail; which is 72 pounds.
5. A person engaged a workman for 48 days. For each day that he labored he received 24 cents, and for each day that he was idle, he paid 12 cents for his board. At the end of the 48 days, the account was settled, when the laborer received $5.04. Required the number of working days, and the number of days he was idle.
Had he worked all the time he would have received 24 X 48=$11.52 ; but he received only $5.04. Therefore by be. ing idle he lost $11.52- $5.04=$6.48. Now for each idle day, he loses the 24 cents, which he might have earned, as well as the 12 cents which he gives for his board ; so that every idle day is to him a loss of 24+12=36 cents. But we have just shown that his total loss was $6.48; ... the num. ber of idle days was 6,48=18; and he worked 48–18= 30 days.
6. A gentleman bought two pieces of silk, which together measured 35 yards. Each of them cost as many shillings per yard as there were yards in the piece, and their whole prices were as 4 to 1. What were the lengths of the pieces ?
SOLUTION. Since each piece cost, per yard, as many shillings as there were yards in its length, it follows that their values expressed in shillings must be as the squares of their lengths. By the question, their prices were as 4 to 1 ; therefore the squares of their lengths must be to each other as 4 to l; consequently, their lengths must be to each other as 2 to 1.
The question is now reduced to the following: Divide 36 into two parts, which shall be to each other as 2 to 1. These parts are 3 of 36 =24, and f of 36=12.
7. In a mixture of wine and cider, 1 of the whole, plus 25 gallons, is wine, and 3 of the whole, minus 5 gallons, is cider. How many gallons were there of each ?
By the question, the wine=1 of the whole +25 gallons, and the cider = $ of the whole - 5 gallons. Hence, taking the sum of these expressions, we get the whole=(1+b) or of the whole + 20 gallons ; • 1 of the whole equals 20 gallons ; consequently, the whole is 120 gallons. ;?.
Now 1 of the whole is 60 gallons, to which, add 25 gal. lons, we get for the wine, 85 gallons.
Again, $ of the whole is 40 gallons, from which, subtracting 5 gallons, we get for the cider, 35 gallons.
8. A market woman bought a certain number of eggs, at 2 for a penny; and as many more, at 3 for a penny; and having sold them again, altogether, at the rate of 5 for 2 pence, found that she had lost 4 pence. How many eggs had she ?
Since, by the question, half of the eggs costs of a penny a piece, and the other half cost $ of a penny a piece, it follows that the average price, which she gave for the eggs, was (+ $)+2=of a penny a piece. Since she sold them altogether at the rate of 5 for 2 pence, that is, of a penny a piece, she must have lost, on each egg, ta-=és of a penny. Therefore, to lose 1 penny, she must dispose of 60 eggs; and to lose 4 pence, she must have had 240 eggs.
9. A and B can, together, do a piece of work in 8 days; A and C.can, together, do it in 9 days. How many days would it require for each to perform the work alone ?