Spring 2009
Statistics 131A
Handout 1
Some important combinatorial objects
1. Factorial notation : For a positive integer r, we use r! to denote the product
r×(r−1) × · · · × 2×1. We also define 0! = 1.
2. Permutations : The number of ways of arranging ndistinct objects in rordered
positions is given by
nPr=n!
(n−r)! =n×(n−1) × · · · × (n−r+ 1), r = 1,2, . . . , n.
3. Binomial coefficients : The number of ways of selecting robjects from a collec-
tion of ndistinct objects :
µn
r¶=n!
r!(n−r)!, r = 0, . . . , n.
4. Binomial theorem :
(x+y)n=
n
X
k=0 µn
k¶xkyn−k.
5. Multinomial coefficients : The number of ways of dividing ndistinct objects
into rgroups of respective sizes n1, . . . , nr, with ni≥0 for all i= 1, . . . , r and
Pr
i=1 ni=n, is given by
µn
n1, n2, . . . , nr¶=n!
n1!n2!· · · nr!.
6. Number of integer solutions : For any positive integer n, number of solutions
of the equation
x1+x2+· · · +xr=n
where x1, . . . , xrare positive integers is given by ¡n−1
r−1¢. If instead the xi’s are only
non-negative, then the number of integer solutions equals ¡n+r−1
r−1¢.
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