...things taken 5 at a time is to the number where 3 are taken at a time as 1 8 is to 5 ; find и. 759. The number of permutations of n things taken r at a time is equal to 10 times the number taken (1 — 1) at a time: and the number of combinations of n things...

...a time. The number of combinations of n things taken r at a time is n(nl)(n-2) (nr + l) " ~E ' For the number of permutations of n things taken r at a time is n(n — 1) (n — 2) (и — r + 1) by Art. 490; and each combination produces |_r permutations, by...

...a time. The number of combinations of n things taken r at a time is n(nl)(n-2) (n-r+ 1) ~Tr " ' For the number of permutations of n things taken r at a time is n(n— l)(n — 2) (n — r+l), by Art. 490; and each combination produces \r_ permutations, by Art....

...additional factor the number of factors is equal to the number of things taken at a time, it follows that the number of permutations of n things taken r at a time is the product of the factors w.(nl)(n— 2) (n—r+}~). ANSWERS. I. (Page 10.) 2. a+3b + 2c. 3. 4. 8«...

...though different permutations, form the same combination, so also do ae and ca, and be and cb. 417. The number of permutations of n things taken r at a time is n(n- l)(n- 2) (n- r + 1). Let there be n letters а, b, с, d, ; we shall first find the number of...

...r &c. n or P= [p [£ [r_ SiO. 186. The number of Combinations of n things taken r at a time is For the number of permutations of n things taken r at a time is and each combination produces V permutations ; hence if Or denote the number of combinations of n things,...

...permutations of n + 1 things ; therefore, &c. See also (233) for the mode of proof by Induction. 95 The number of permutations of n things taken r at a time is denoted by P (n, r). P (n, r) = n(nl)(n-2) ... (n—r+ï) = пм. By (94) ; for (те— г) things...

...taken at a time, there will be r factors, and the last one will be n - (r - 1) or n - r + 1. Therefore the number of permutations of n things, taken r at a time, is n(n - l)(n - 2) .... to r factors, or, denoting the number by Pr, we have 229. Corollary. — When...

...principal theorem in Permutations, to which reference is made in the note on page 289. To 'prove that the number of permutations of n things taken r at a time is n. (n-1) (n — r+1). Let there he n things a, b, c, d If n things be taken 1 at a time, the number...

...and x*. 10. Enunciate the binomial theorem, and prove it for a positive fractional exponent. 11. If the number of permutations of n things taken r at a time is equal to 10 times the number of permutations of n things taken r—1 at a time, find r. 12. Find the...