HP 10s Instruction Manual page 2

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HP 10s Probability – Rearranging Items
Rearranging items
The calculation of probabilities often involves counting the ways in which items can be rearranged. We have three
functions at our disposal for this purpose: factorial, permutations and combinations.
If n is a positive integer, its factorial (whose symbol is
0! is defined as 1.
A permutation of r from n is a way in which a set of r elements may be chosen in order from a set of n elements. In other
words, it's an ordered subset of a set of distinct objects. The number of possible permutations, each containing r objects,
that can be formed from a collection of n distinct objects is given by:
The above expression also shows the multiple ways of symbolizing permutations. When a permutation involves all the
elements of a set (i.e. r = n) then it is called a rearrangement, and also as shuffle especially when cards are used. Notice
that in this case:
A combination is a selection of one or more of a set of distinct objects without regard to order. These are the different
ways of denoting combinations, and the number of possible combinations, each containing r objects, that can be formed
from a collection of n distinct objects:
As far as rearranging items is concerned, whether order is important or not is the only difference between combinations
and permutations.
!
One of the most important theorems on combinations is Vandermonde's theorem, which states that:
The factorial function was traditionally used for calculating permutations and combinations, which show up in many
discrete probability distribution calculations, such as the binomial and hypergeometric distributions.
On the HP 10s all these functions are executed using the b, F and c keys. The HP 10s errors for factorials of
integers greater than 69, but this covers most of the problems you will probably find, although numbers larger than 69
can be used for combinations and permutations, if the result does not cause an overflow.
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! n n
= !
n n
P = P
=
r r
!
n n
C = C
=
r r n
! n
) is the product of all the positive integers up to and including n:
n ( ) n ( ) ...
1 2
" ! " ! !
P = P(n,r) = (n)
=
n r r
! n
n n
P P ! n
= = =
r n
!
0
n
C = C(n,r) = ( ) =
r r
(n " r)!r!
n n n
1 1
! !
C C C
= +
r r 1 r
!
- 2 -
2 1
!
n!
(n " r)!
n!
HP 10s Probability – Rearranging Items - Version 1.0