...are ab, ac, be. The Combination of a, b, and c, taken three or all together, is one, viz. abc. 180. To find the number of combinations of n different things taken r together. Let C, denote the number required. Now since every combination of r letters admits of (1.2...

...kind), q all alike (of another kind), r all alike (of another kind), and the rest all different, VI. The number of combinations of n different things, taken r at a time, _n(nl) (n-2}...(n- r+l) \rfn - r ' VII. The total number of combinations of n different things = 2"-l....

...together when / of them are of one kind, q of another, r of another, and the rest all different \n 80. The number of combinations of n different things, taken r at a time . . _ n(n—i)(n — 2) .... (n — r+i) ' ----- ' 17 | r | n— r 81. The total number of combinations...

...from each combination we could make six permutations ; thus abc, acb, bac, bca, cab, cba : and so on. And, generally, in accordance with Art. 405, any combination...permutations. 408. To find the number of combinations ofn. different things taken r at a time. Let C, denote the number of combinations required. Since each...

...bed. The number of combinations of n different things r at a time is denoted by the symbol KGr. 240. To find the number of combinations of n different things taken r at a time. Let the different things be represented by the letters a, b, c... Now in the combinations of the n letters...

...in which 6 persons can be seated at a dinner table. Ans. 720. 181. The Number of Combinations. — To find the number of combinations of n different things taken r at a time. The number of permutations of n things taken r at a time is n(n - l)(n - 2) ....... (n - r + 1). (Art....

...time is equal to the product of the natural numbers from 1 to n inclusive. (See Note, Art. 444.) 544. To find the number of combinations of n different things taken r at a time. The number of permutations of n different things taken r at a time, is n(n-1)(n-2)...(w-r+l) (Art....

...must be even, and r must be equal to \n . 376. Number of Combinations of n things taken r together. The number of combinations of n different things taken r at a time is given by the formula First Proof. Each of the combinations of n things taken r at a time contains...

...many permutations as combinations (Art. 200), we have ,C, X _r = „Pr (2) Hence, from (1) and (2) The number of combinations of n different things, taken r at a time, is Multiplying both terms of the second member of (3) by the product of the natural numbers from 1...

...term of an AP is — 2, and the sum of the first 8 terms is — 10, find the nth term. 9. Show that the number of combinations of n different things taken r at a time can be found independently of the number of permutations. 10. (a) Deduce the formula called the Binomial...