Also, q things may be disposed of in q+1 ways, and r things in r +1 ways, and so on. Hence, the number of ways in which all may be disposed of is (p + 1)(q + 1)(r + 1)....... This includes the case in which all are rejected, which is inadmissible. (1) In how many ways can we select, one or all, from 5 arithmetics, 3 readers, and 4 spellers? The arithmetics may be disposed of in (p+1) ways, or 6 ways. The readers in (q + 1), or 4 ways. The spellers in (r+ 1), or 5 ways. Hence, 6.4.5-1=119. 429. XIII. Permutations and combinations of n things, not all different, taken r at a time. EXAMPLES: (1) How many combinations can be made from 6 flags, 4 white and 2 black, taken 4 at a time? 1. How many permutations can be made with the letters of the word Constantinople? 2. How many words, consisting of 4 consonants and 2 vowels, can be formed from 10 consonants and 5 vowels? 3. There are 7 candidates for a classical, 5 for a mathematical, and 4 for a divinity scholarship, in how many ways can the scholarships be awarded? 4. How many permutations can be made from the letters. in the word Ecclesiastical? 5. With 4 consonants and 3 vowels, how many words of 5 letters can be formed? 6. With 6 consonants and 4 vowels, how many words of 4 letters each can be formed, if each begins with a consonant and ends with a vowel? 7. Out of 30 soldiers, in how many ways can 6 officers be chosen, and how many times would any man be chosen? 8. How many words of 5 letters can be made from the letters of the word triangle, if all end with the letter e? 9. Find the value of 12P, 12C4 10. Find the number of combinations of 75 things taken 70 at a time. 11. How many changes can be rung with 7 bells? 12. In how many ways can a football eleven be chosen from 39 men, the captain always being one? 13. Out of 25 candidates for a baseball nine, in how many ways can the three bases be filled? 14. How many different throws can be made with five dice, if a six always turns up? 15. How many ways can the men of an eight-oared crew be arranged in the boat, if the captain always rows stroke? 16. At a whist table, what are the chances that A will have all clubs, B all spades, Call hearts, D all diamonds? Give formula only. 17. Out of 20 consonants and 5 vowels, how many words can be formed consisting of 4 consonants and 3 vowels? 18. How many different committees can be formed from 16 Republicans and 10 Democrats, if each committee consists of 4 Republicans and 3 Democrats? 19. How many ways can a crew of eight be arranged, if two always row on starboard and one on port? 20. How many signals can be made with eight flags of different colour, four at a time? 21. How many different signals can be made with 10 lights of different colour, 3 at a time? 22. How many signals can be made with 12 flags, of which 6 are red, 3 are white, 2 are blue, and 1 is black, if they may be hoisted one above the other, side by side, or diagonally? 23. Out of 10 physicians, 8 lawyers, and 12 ministers, how many committees can be formed consisting of 3 physicians, 2 lawyers, and 4 ministers? 24. In how many ways can 25 tutors be assigned to watch 3 examinations, 3 to the first, 4 to the second, and 5 to the third? CHAPTER XXXV. INTEREST AND ANNUITIES. 430. I. To find the simple interest or amount of a given sum for a given time. 431. The Principal is the sum lent. The Interest is the money paid for what is lent. The Amount is the sum of the principal and interest at the time the money is due. Simple Interest is paid on the principal alone. Let Hence, P represent the principal. r the interest on $1 for 1 year. n the number of years. R the amount of $1 for 1 year. R=1+r. Pr Interest on P for 1 year. Interest on P for n years. P(1 + nr) = Amount of P for n years. That is, P(1+ nr) = A. When interest is mentioned, simple interest is meant. (1) When three of the numbers P, n, r, A are given, the fourth can be found. EXAMPLE: Find the number of years in which $250 will amount to $500 at 6%. Here A, P, and n are given, and r required. 432. II. To find the interest or amount of a given sum at compound interest for a given time. (I.) When payable annually. P(1+r) or PR = Amount for 1 year. PR (1+r) or PR2 = Amount for 2 years. PR"-1 (1+r) or PR" Amount for n years. That is, = A = PR". 433. (II.) When payable semiannually. (2) In like manner we may show that when the interest is payable quarterly or monthly, that |