Question

(a) To determine the singless String numbers of the first kind of \(C(3,2)\).

(b) To determine the singless String numbers of the first kind of \(C(4,2)\).

(c) To determine the singless String numbers of the first kind of \(C(4,3)\).

(d) To determine the singless String numbers of the first kind of \(C(5,2)\).

Short answer

(a) The singless String numbers of the first kind of\(C(3,2)\)is 3 ways.

(b)The singless String numbers of the first kind of\(C(4,2)\)is 11 ways.

(c)The singless String numbers of the first kind of\(C(4,3)\)is 6 ways.

(d) The singless String numbers of the first kind of \(C(5,2)\) is 10 ways

Step-by-step solution

 Given

(a) The first kind of \(C(3,2)\).

(b) The first kind of \(C(4,2)\).

(c) The first kind of \(C(4,3)\).

(d) The first kind of \(C(5,2)\).

The Concept of Stirling numbers formula

The objects are labelled but the boxes are unlabelled, we can use the Stirling numbers formula

\(\sum\limits_{j = 1}^k {\frac{1}{{j!}}} \sum\limits_{i = 0}^{j - 1} {{{( - 1)}^i}} C(ji){(j - i)^n}\)

The singless String numbers of the first kind (a)

\(C(3,2)\)means when 3 people seated around 2 empty tables, the one member has to seat alone and other two members has to sit together, the lone member can be chosen in 3 ways, where as other two members can be chosen around round table is one way.

Hence

The singless String numbers of the first kind of \(C(3,2)\) is 3.

The singless String numbers of the first kind (b)

\(C(4,2)\)when 4 members want to be seat around two tables,

There are two possibilities, either two members can seat in each table or the three members can seat in a table and one member can seat alone.

In first possibility then there are three ways to decide who sits with member A,

In second possibility alone member can choose by 4 ways and then two ways to arrange other two members.

The singless String numbers of the first kind of \(C(4,2) = 3 + 4.2 = 11\) ways.

The singless String numbers of the first kind (c)

\(C(4,3)\)when 4 members want to be seat around 3 tables.

There is only one possibility, that is two members sitting alone and one pair sitting together.

The number of ways to select 2 members out of 4 is 6

The singless String numbers of the first kind of \(C(4,3) = 6\) ways.

The singless String numbers of the first kind (d)

\(C(5,2)\)when 5 members want to be seat around two tables,

In this case also one possibility, the three members' seat alone and other pair sitting together,

Thus selecting 3 persons from out 5 is 10 ways.

Hence, the singles String numbers of the first kind of \(C(5,2) = 10\) ways.