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330 ALGEBRA. 3. How many words can be formed from the letters of the word Simoom, so that vowels and consonants occur alternately in each word? 4. A telegraph has 5 arms, and each arm has 4 distinct positions, including the position of rest: find the total number of signals that can be made. 5. In how many ways can n things be given to m persons, when there is no restriction as to the number of things each may receive? 6. How many different arrangements can be made out of the letters of the expression $$a^5b^3c^6$$ when written at full length? 7. There are 4 copies each of 3 different volumes; find the number of ways in which they can be arranged on one shelf. 8. In how many ways can 6 persons form a ring? Find the number of ways in which 4 gentlemen and 4 ladies can sit at a round table so that no two gentlemen sit together. 9. In how many ways can a word of 4 letters be made out of the letters $$a, b, e, c, d, o,$$ when there is no restriction as to the number of times a letter is repeated in each word? 10. How many arrangements can be made out of the letters of the word Toulouse, so that the consonants occupy the first, fourth, and seventh places? 11. A boat's crew consists of eight men of whom one can only row on bow side and one only on stroke side: in how many ways can the crew be arranged? 12. Show that n+1Cr = nCr + nCr-1. 13. If 2nC3:nC2=44:3, find n. 14. Out of the letters $$A, B, C, p, q, r$$, how many arrangements can be made beginning with a capital? 15. Find the number of combinations of 50 things 46 at a time. 16. If 18Cr = 18Cr+2, find rC5. 17. In how many ways is it possible to draw a sum of money from a bag containing a dollar, a half-dollar, a quarter, a dime, a five-cent piece, a two-cent piece, and a penny ?