1.SampleSpaceandProbabilityPartV:Coun8ng‐2
ECE302Fall2009TR3‐4:15pmPurdueUniversity,SchoolofECE
Prof.IlyaPollak
Par88ons
• Howmanywaysaretheretodividenitemsintorgroupssothatthei‐thgrouphasniitems(wheren1+n2+…+nr=n)?
• Example:ifr=2,theansweris
€
nn1
ways of forming the first group
n − n1 objects remain
n − n1
n2
ways of forming the second group
n − n2 objects remain
€
Answer : nn1
n − n1
n2
⋅… ⋅
n − n1 −…− nr−1
nr
=n!
n1!(n − n1)!⋅
(n − n1)!n2!(n − n1 − n2)!
⋅… ⋅(n − n1 −…− nr−1)!
nr!(n − n1 −…− nr−1 − nr )!= 0!=1
=n!
n1!n2!…nr!
These are denoted n
n1,n2,…,nr
and called multinomial coefficients.
The reason for this name is that
x1 + x2 +…+ xr( )n =n
n1,n2,…,nr
n1 +n2 +…+nr = n∑ x1
n1 x2n2 ⋅… ⋅ xr
nr
Example(p.68)• 90students,includingJoeandJane,aretobesplitinto3
classesofequalsize,atrandom.
• WhatistheprobabilitythatJoeandJaneendupinthesameclass?
Example• 90students,includingJoeandJane,aretobesplitinto3
classesofequalsize,atrandom.
• WhatistheprobabilitythatJoeandJaneendupinthesameclass?
Example• 90students,includingJoeandJane,aretobesplitinto3
classesofequalsize,atrandom.
• WhatistheprobabilitythatJoeandJaneendupinthesameclass?
Example• 90students,includingJoeandJane,aretobesplitinto3
classesofequalsize,atrandom.
• WhatistheprobabilitythatJoeandJaneendupinthesameclass?
Asimplermethod
• PlaceJoeinoneclass.• ForJane,thereare89totalpossibili8es,ofwhich29putherinthesameclasswithJoe.