| Factorial | ! 0 0 0 | Out Of (Combinations) |
For a non-negative integer argument y, the definition
is */>:i.y . In general, !y
is G(1+y) (the gamma function).
Thus:
(*/1 2 3 4 5) , (!5)
120 120
]x=: 2 %~ 3 -~ i. 2 4
_1.5 _1 _0.5 0
0.5 1 1.5 2
!x
_3.54491 _ 1.77245 1
0.886227 1 1.32934 2
]fi=:!^:_1(24 25 2.1 9876)
4 4.02705 2.05229 7.33019
! fi
24 25 2.1 9876
|
For non-negative arguments x!y is the number of ways
that x things can be chosen out of y .
More generally, x!y is (!y)%(!x)*(!y-x) . Thus:3!5 10 (!5)%(!3)*(!5-3) 10 1j2 ! 3.5 8.64269j16.9189 ]y=:2&!^:_1 (45 4.1 30 123) 10 3.40689 8.26209 16.1924 2 ! y 45 4.1 30 123 ]x=:!&10^:_1 (2.5 45) 0.3433618 2 x ! 10 2.5 45 |
h=: 0,i=: i.5 [ j=: -1+i.5 [ k=: 5#1
tables=: (,.h);(i,i!/i);(j,i!/j);(k,i(+/\^:)k)
format=: ({. ,:&< }.)@":&.>
format tables
+---+-----------+-------------------+--------------+
|+-+|+---------+|+-----------------+|+------------+|
||0|||0 1 2 3 4|||_1 _2 _3 _4 _5|||1 1 1 1 1||
|+-+|+---------+|+-----------------+|+------------+|
||0|||1 1 1 1 1||| 1 1 1 1 1|||1 1 1 1 1||
||1|||0 1 2 3 4|||_1 _2 _3 _4 _5|||1 2 3 4 5||
||2|||0 0 1 3 6||| 1 3 6 10 15|||1 3 6 10 15||
||3|||0 0 0 1 4|||_1 _4 _10 _20 _35|||1 4 10 20 35||
||4|||0 0 0 0 1||| 1 5 15 35 70|||1 5 15 35 70||
|+-+|+---------+|+-----------------+|+------------+|
+---+-----------+-------------------+--------------+
Figurate numbers of order zero are all ones; those of higher
orders result from successive applications of subtotals
(that is, sums over prefixes, or +/\). Those of order
two are the triangular numbers, resulting from subtotals
over the integers beginning with one.