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PERMUTATIONS AND COMBINATIONS. 325 4. How many different selections of four coins can be made from a bag containing a dollar, a half-dollar, a quarter, a florin, a shilling, a franc, a dime, a sixpence, and a penny? 5. How many numbers between 3000 and 4000 can be made with the digits 9, 3, 4, 6? 6. In how many ways can the letters of the word volume be arranged if the vowels can only occupy the even places? 7. If the number of permutations of n things four at a time is fourteen times the number of permutations of n-2 things three at a time, find n. 8. From 5 teachers and 10 boys how many committees can be selected containing 3 teachers and 6 boys? 9. If 20Cr = 20Cr-10, find rC12, 18Cr. 10. Out of the twenty-six letters of the alphabet in how many ways can a word be made consisting of five different letters two of which must be a and e? 11. How many words can be formed by taking 3 consonants and 2 vowels from an alphabet containing 21 consonants and 5 vowels? 12. A stage will accommodate 5 passengers on each side: in how many ways can 10 persons take their seats when two of them remain always upon one side and a third upon the other?

397. Hitherto, in the formulae we have proved, the things have been regarded as unlike. Before considering cases in which some one or more sets of things may be like, it is necessary to point out exactly in what sense the words like and unlike are used. When we speak of things being dissimilar, different, unlike, we imply that the things are visibly unlike, so as to be easily distinguishable from each other. On the other hand, we shall always use the term like things to denote such as are alike to the eye and cannot be distinguished from each other. For instance, in Ex. 2, Art. 396, the consonants and the vowels may be said each to consist of a group of things united by a common characteristic, and thus in a certain sense to be of the same kind; but they cannot be regarded as like things, because there is an individuality existing among the things of each group which makes them easily distinguishable from each other. Hence, in the final stage of the example we considered each