Taken two at a time:{ae, ai, ao, au, ei, eo, eu, io, iu, ou} or 10 combinations.
Taken four at a time {aeio, aeiu, aeou, aiou, eiou} or 5 combinations.
The number of combinations C is given by
)!(!
!
rNr
N
C
N
r
−
=
where N! = N×((N–1)×(N–2)× … × 2 × 1. To calculate this using 42S just enter N, press
ENTER, enter r and press PROB COMB.
PERM: This calculates the number of permutations of N things taken r at a time (mathematically
notated as P
r
N
). A thing cannot appear more than one time but now the order matters.
Example: Five cars are in a race. Their colors are red, blue, green, white and cyan. What are
the possible results for the first, second, and third place winners?
Solution: For the first position we have five possibilities. For the second position we have
four possibilities, and three possibilities for the third position. So we have 5x4x3=60 different
arrangements. To see this using 42S just enter 5, press ENTER, enter 3 and press PROB
PERM. It is simple to realize that the number of permutations is given by
)!(
!
rN
N
P
N
r
−
=
In particular if r = N (all the things are taken) then the number of permutation is N!.
Example: In how many ways we can re-arrange the letters of the word “love”.
Solution: 4!=24.
N!: This just calculates the factorial of N given by N!=N.(N-1)...1 for a number (non-negative
integer). The biggest number allowed is HP-42S is 253 and in Free42 is 170.
GAM: This is the Gamma function which is defined by
∫
∞
−
=Γ
0
1
)( dxexa
xa
For a integer number we have Γ(n)=(n–1)! and Γ(n+1)=n! but the argument of the gamma
function can be a non-integer (but must be real). In this point HP-42S is different from the
33S which has only one function for both things.
RAN: This is the random number generator which gives a pseudo-random number in 0 ≤ x ≤ 1.
SEED: A sequence of pseudo-random numbers always starts with a seed. If you repeat the seed the
sequence repeats. To enter a new seed just enter a number and press SEED. If the seed is zero
the calculator will generate another seed.
5 Complex Numbers
5.1 Complex numbers in rectangular coordinates.
Unlike the HP-33S (and its ancestor HP-32SII) complex numbers are straightforwardly supported and
used in the HP-42S. There is almost nothing special to say. Just enter –1 and press √x, what are you
going to have is x: 0.0000 i1.0000 which means i. (Just for comparison, to do the same in HP-33S we
have to do 0 ENTER 1 +/– ENTER 0 ENTER .5 CMPLX y
x
and we will have 0 and 1 meaning i)
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