Generalization of Some Problems with s-Separation

In this article we apply and discuss El-Desouky technique to derive a generalization of the problem of selecting k balls from an n-line with no two adjacent balls being s-separation. We solve the problem in which the separation of the adjacent elements is not having odd and even separation. Also we enumerate the number of ways of selecting k objects from n-line objects with no two adjacent being of separations m, m + 1, ···, pm, where p is positive integer. Moreover we discuss some applications on these problems.


Introduction
Kaplansky [1] (see also Riordan ([2] p. 198, lemma) and Moser [3]) studied the problem of selecting k objects from n objects arranged in a line (called n-line) or a circle (called n-circle) with no two selected objects being consecutive.Let ( ) , f x y and ( ) , g x y denote the number of ways of such selections for n-line and n-circle respectively.Kaplansky proved that ( ) (1.1) and ( ) .
El-Desouky [4] studied another related problem with different techniques and proved that ( ) where ( ) , l n k is the number of ways of selecting k balls from n balls arranged in a line with no two adjacent balls being unit separation.
In the following we adopt some conventions: ( ) denotes the coefficient of n x in the formal power series ( ) ( ) x y in the series ( ) x is the largest integer less than or equal to x, Also, El-Desouky [5] derived a generalization of the problem given in [4] as follows: let ( ) N where the dif- ference 1 s + is not allowed, so where min 1, , 0 , and 0,1, , .

Main Results
We use El-Desouky technique to solve two problems in the linear case, with new restrictions.That is if the separation of any two adjacent elements from the k selected elements being of odd separation and of even separation.Moreover, we enumerate ( ) , ; , s M n k m pm which denotes the number of ways of selecting k objects from n objects arrayed in a line where any two adjacent objects from the k selected objects are not being of m, m + 1, •••, pm separations, where p is positive integer.

( )
, o y n k denote the number of ways of selecting k balls from n balls arranged in a line, where the separa-tion of any two adjacent balls from the k selected balls being of odd separation.say s, i.e.
, where , where Therefore, the coefficient of n x gives ( ) ( ) Moreover in the next subsection, we use our technique to enumerate ( ) , ; , s M n k m pm the number of ways of selecting k objects from n objects arrayed in a line such that no two adjacent elements have the differences m + 1, m + 2, •••, pm + 1 i.e. no two adjacent element being of m, m + 1, •••, pm separations, where p is positive integer., ; ,

Explicit Formula for
, where

Some Applications
Let n urns be set out along a line, that is, one-dimensional.
as a special case of El-Desouky results [5].
It is of practical interest to find the asymptotic behavior of ( ) , f n k or the probability ( ) ( ) Let X be a random variable having the probability function ( )

s l n k denote
the number of ways of selecting k balls from n balls arranged in a line with no two adjacent balls from the k selected balls being s-separation; two balls have separation s if they are separated by exactly s balls.Let denote the number of ways of selecting k balls from n balls arranged in a circle with no two adjacent balls from the k selected balls being s-separation Let ( ) , s l n k be as defined before.Then ( ) , s l n k is equal to the number of k-subsets of n

s l n k
m be the number of ways of selecting k balls from n balls arranged in a line with exactly m adjacent balls being of separation s or ( )-successions s m pm be the number of ways of selecting k objects from n objects arrayed in a line where any two adjacent objects from the k selected objects are not being of m, m + 1, •••, pm separations, where p is positive integer, hence

Table 1 .
table for the values of A calculated table for the values of denote the number of ways of selecting k balls from n balls arranged in a line, where the separation of any two adjacent balls from the k selected balls are not being of even separation, say s i.e.
o y n k is given in Table1, where 1 n