... q - permutations and the numbers of q - combinations , without and with repetitions . Lemma 1.1 ( a ) The number of q - permutations of r out of n equals [ n ] r , q , while the number of q - permutations of r out of n , with repetition ...
... Q , 199 = :, 100 A x B ( Cartesian product ) , 126 A ( arrangements without repetition ) , 329 Ch ( combinations ... Q ( permutations with repetition ) , 337 k 1 R ° −1 ( inverse relation ) , 132 R ( arrangements with repetition ) , 330 yx ...
... q Combinations with repetitions Suppose we are to place r balls of the same colour in n numbered boxes allowing as many balls as we wish. Then the number of ways to place the balls is ( 1)! n r !()! rnr = C(n + r 1, r) Consider ...
... q combinations from one of them , and p.q combinations from all of them . in ... repetition ) be made of them , including their rth powers and the products ... combinations of r letters , without repetitions , of ( n + r - 1 ) letters ...
... q letters $ p + p of which are alike . 1st . With the q unlike letters we can form C , r - com- binations . 2nd . Taking one of the p letters , and r - 1 of the q , we ... COMBINATIONS WITH REPETITION 11 In the 10 CH . XXIII P LETTERS ALIKE.
... combinations exist of the letters P , Q , R , S , T , U , the condition being that some one of these letters shall ... repetition ; as before shown . This is the most simple proof of the formula for combinations with repetition . n = In ...
... Q.'s version . There the Erinyes are called to witness Ajax's death at the ... combination , with adversative force , in Q. see Vian and Battegay s.v. ye III ; it ... repetition within the line as first word of the next sentence expresses ...
... combinations of numbers 1 to m taking q - 1 at a time and the number of combinations q - 1 at a time when each number is repeated ... repetition when two or more letters are repeated one or more times . Each combination determines the literal ...
... q,,, = qum derive the symbolic expression qm = (1 + E + E2/2)'"0!, (EkO! = k!) and the recurrence relation qm = — 1)qm-1 _ m(m _ DIM-2 + m + 1 Verify that qO =1, q1 = 3, q2 =19, q3 = 271. 14. By classifying the permutations with repetition ...
... repetition , have the combination factor , Q , = and the permutation factor ... Q x P therefore coincides with the expression given by Lagrange , in his demon- stration of the Polynome theorem , for the ... COMBINATIONS AND PERMUTATIONS . 487.