Principles of Mathematics in Operations Research

208 Solutions

an urn. Let us fix a ball, call it super ball. Two mutually exclusive alternatives

exist; we either select the super ball or it stays in the urn. Given that the

super ball is selected, the number of different ways of choosing r — 1 balls out

of n— 1 is (™l|). In the case that the super ball is not selected, (n~^1 ) denotes

the number of ways of choosing r balls out of n — 1. By the rule of sum, the

right hand side is equal to the left hand side.

(«0(S) + G) + -+O=2»:

We will use the corollary to the following theorem.

Theorem S.l.l (Binomial Theorem)

(i+.)-=cy+cy+...+(^

Corollary S.1.2 Let x = 1 in the Binomial Theorem. Then,

Combinatorial Method:

2" is the number of subsets of a set of size n. (JJ) = 1 is for the empty set,

(^) = 1 is for the set itself, and ("), r = 2,..., n — 1 is the number of proper

subsets of size r.

way (?) = (") (r_;)=

Forward - Backward Method:

n\ fm\ n\ m\ n\

mj \r J (n — m)\ ml (m — r)\r\ (n — m)\ (m — r)\r\

n\fn — r\ n\ (n — r)\ n\

r) \m — T) r\ (n — r)! (n — m)! (m — r)! r! (n — m)\ (m — r)\

Combinatorial Method:
(^) denotes the number of different ways of selecting m Industrial Engineering
students out of n M.E.T.U. students and (™) denotes the number of different
ways of selecting r Industrial Engineering students taking the Mathematics
for O.R. course out of m I.E. students. On the other hand, (™) denotes the
number of ways of selecting r Industrial Engineering students taking Math-
ematics for O.R. from among n M.E.T.U. students and (^1^.) denotes the
number of different ways of selecting in — r Industrial Engineering students