Mathematical Questions and Solutions Volume 43 by Books Group
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Author: Books Group
Page Count: 38 pages
Published Date: 20 May 2012
Publisher: Not Avail
Publication Country: United States
Language: English
ISBN: 9781236326911
File size: 16 Mb
Download Link: Mathematical Questions and Solutions Volume 43
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1885 Excerpt: ...to find the locus of a point at which the second set could be put in simple perspective." There is, of course, a unique point in the plane at which they are projectives (see Salmon's Higher Curves, 330, and Townsend's Modern Geometry, Vol. II., p. 336. 7939. (By H. Ll. Smith, M.A.)--A district containing 2n Liberal and Conservative voters is divided into three equal wards, each returning one member. Show that, if n be odd, the chance of one Conservative being returned is 3 ( + 3) /4 ( + 2). Solution by the Proposer. Consider one of the wards. It may contain 0, 1, 2, ... n Conservatives. And number of ways in which it may contain r Conservatives = number of ways in which the remaining (n--r) may be divided between the other two wards = ( --r+ 1); hence the number of ways in which the Conservatives can be distributed in the first ward=2JH ( --r + 1) =i ( + 1)( + 2); and the number of ways in which they may be in a majority is 2r(.+1)( -'-+l)-( + l)( + 3); hence the chance of a Conservative being returned in the given ward is "+-;therefore the chance of a Conservative being returned in one of the 4 ( + 2) b three wards is HLtQ. 4( + 2) giving (a'-bx-2 = a + 46 +-3a34V, and similarly for y and z. (See Vol. Xlii., p. 43.) At the end of the Appendix to Vol. Xlii., there is another solution of this question, which, though every equation is correct, arrives at the false conclusion that x, y, z are variable. The error, however, nowise invalidates the method which the solution was intended to illustrate. For, substituting in a"y + bh + A: = 0 the values y = x--(a2--i2) / S, z = x--(a2--c2) / S, we at once obtain x 1 _ y _ by symmetry; after which the solution may be compl...
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