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Sinossi
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. 1876 Excerpt: ...i--1 together will be (re-l)(re-2)(re-3... (re-l)-(r-l) + l, or (re-l)(re-2)(re-3)...(re-r+l). We can place a before each of these, and we shall have this as the number of permutations of n things taken r together, in all of which a stands first. The same will be true for b, c,d...; hence we see that the permutations of re things taken r together are re(re-l) (re-2)(re-3)... (re-r+1). This result shews us that if this expression be correct when the quantities are taken »--1 together, it is also true when they are taken r together, i. e. for the next higher number. But we have proved it to be true when r=2, or 3; hence it is true when r=4, and therefore when r = 5, and so on: that is, it is generally true whatever value r may hare. Ex. 1. In how many ways can a four-oar be arranged, if there are 7 rowers to choose from? Here there are the permutations of 7 rowers taken 4 together j hence we have 7x6x5x4 = 840. Ans. 157. The number of Permutations of a things taken all together tn(n-l)(n--2)... 3. 2.1. In the last Art. let r = n, thus the expression becomes W(«-1)(w-2)...(w-w+1)=m(«-1)(»-2)... 1. This is frequently written in the form n and is read factorial n. Thus 5 stands for 1 x 2 x 3 x 4 x 5. Ex. 2. In how many ways can 6 persons be arranged at a dinner-table? Ans. = 6 x 5 x 4 x 3 x 2 x 1 = 720. 158. The number of Permutations of n things in which a recurs p times, b recurs q times, c recurs r times, &c. is 1 p.q.r... Let N represent the required number of permutations. Now suppose that in each permutation all the letters a were changed into different letters; since there are p of them, these new letters would form p times as many permutations as there were before (Art. 157); i.e. the number of permutations would be Nx p. Similarl...
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