Principles of Mathematics in Operations Research

Solutions 287

(x + y)n = Yf(^)(x 1 + -.. + xl)ixti

i=0

. „ ,t/ *-^ \«i,••-,«/

i\-\ Yi\ — i

Y^ I • ". • ix^-x^x

(^71) »i «i™«'+i
i x; x+i
b) For each object i = 1,..., n, either it is not selected or selected once, twice,
thrice, and so on; that is x, £ Si = Z+. Then,
n
</(x) = JJ^^0 + x^1 + x^2 + • • •) = (1 + x + x^2 + • • •)",
»=i
Without loss of generality, we may assume that r = J3 Xj objects are selected.
We know from 14.4 that the number of distinct ways of selecting r objects
out of n objects with replacement is ("j^^1 ) = (n~"+r). Thus, ar = (n~l+r).
Therefore,
5 (x) = (i+x+x
2

• --.)
  n
  = ]r(
  n
  ~
  1+r
  V.
  r=0 ^ '
  xi + x 2 + x 3 + x 4 = 13, Xi = 1,2,3,4,5,6 Vi =>
  5 (x) = (x + x
  (^2) + x (^3) + x (^4) + x (^5) + x (^6) ) (^4) = x (^4) (l + x + x (^2) + x (^3) + x (^4) + x (^5) ) 4
  We are interested in the coefficient of x^13 of g(x), which is the coefficient of
  x^9 of h(x) = (1 + x + x^2 + x^3 + x^4 + x^5 )^4.
  p(x) = 1+ x + x^2 + x^3 + x^4 H
  xp(x) = x + x^2 + x^3 + x^4 + • • •
  (1 - x)p(x) — 1 =^> p(x) =
  1-x