By Pei J., Zhang X.

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Two & s is 115 divided by 113 and 112. are in fact The arrangements bbaaa, babaa, baaba, baadb abbaa, ababa, abaab aabba, aabab aaabb The first line has a& at the beginning, and there are four positions for the second b ; the next line has a b in the second place, and there are three new positions for the other b, rived at the We and so on. might of course have ar number of arrangements in this particular case by the far simpler process of direct counting, which we have used as a verification *; but the advantage D of our longer process is that it will give us a general formula applicable to all cases whatever.

May be got from c 2 THE COMMON SENSE OF THE EXACT SCIENCES. 20 Now this is quite completely and satisfactorily nevertheless we are going to prove it again in proved another way. ; The reason the proposition is further still ; we want to extend we want to find an ex that pression not only for the square of (a + fy, but for any other power of it, in terms of the powers and products of powers of a and b. And for this purpose the of proof we have hitherto adopted is unsuitable. mode We cube of a-t-b by adding the proper pieces to the cube of a ; but this would be some what cumbrous, while for higher powers no such repre might, it is true, find the The proof to which we now pro ceed depends on the distributive law of multiplication.

Muting the capital letters we can make lip arrange jot the small letters ments, and by permuting the small letters Hence every arrangement rangements. and smalls Uq ar in respect of one of a group of Tip x Yiq Now the whole number of equivalent arrangements. capitals is arrangements of the p + q letters is IT (p + q) and, as we have seen, every arrangement in respect of capitals ; and smalls is here repeated Yip x Tlq times. Conse we are in search of is got by di quently the number viding IT (p + q) by lip x Uq.

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1-Generator Quasi-Cyclic Codes by Pei J., Zhang X.

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