Theorem 2.3 (page 44)
The number of ways of partitioning n distinct objects into k distinct groups
containing 𝑛1, 𝑛2, … , 𝑛𝑘 objects, respectively, where each object appears in
exactly one group and ∑𝑘𝑖=1 𝑛𝑖 = 𝑛, is
𝑛!
𝑛
𝑁 = (𝑛 𝑛 … 𝑛 ) =
1 2
𝑘
𝑛1 ! 𝑛2 ! ∙∙∙ 𝑛𝑘 !
Example 2.10 (page 45)
A labor dispute has arisen concerning the distribution of 20 laborers to four
different construction jobs. The first job (considered to be very undesirable)
required 6 laborers; the second, third, and fourth utilized 4, 5, and 5 laborers,
respectively. The dispute arose over an alleged random distribution of the
laborers to the jobs that placed all 4 members of particular ethnic group
on job 1. In considering whether the assignment represented injustice, a
mediation panel desired the probability of the observed event. Determine
the number of sample points in the sample space S for this experiment. That
is, determine the number of ways the 20 laborers can be divided into groups
of the appropriate size to fill all of jobs. Find the probability of the observed
event if it is assumed that the laborers are randomly assigned to jobs.
Solution:
20!
20
𝑁= (
)=
6455
6! 4! 5! 5!
16!
16
𝑛𝑎 = (
)=
2455
2! 4! 5! 5!
𝑃(𝐴) =
𝑛𝑎
= 0.0031
𝑁
Since the probability is very small, the jobs were nor randomly assigned.