Twelvefold way
Injective functions from N to X

This case is equivalent to counting sequences of n distinct elements of X, also called n -permutations of X, or sequences without repetitions; again this sequence is formed by the n images of the elements of N. This case differs from the one of unrestricted sequences in that there is one choice fewer for the second element, two fewer for the third element, and so on. Therefore instead of by an ordinary power of x, the value is given by a falling factorial power of x, in which each successive factor is one fewer than the previous one. The formula is

Note that if n > x then one obtains a factor zero, so in this case there are no injective functions N → X at all; this is just a restatement of the pigeonhole principle.

Example:

X = { a, b, c, d }, N = { 1, 2 }, then {\displaystyle X=\{a,b,c,d\},N=\{1,2\}{\text{, then }}}

| { (a, b), (a, c), (a, d), (b, a), (b, c), (b, d), (c, a), (c, b), (c, d), (d, a), (d, b), (d, c) } | = 4 2 _ = 4 × 3 = 12 {\displaystyle \left\vert \{(a,b),(a,c),(a,d),(b,a),(b,c),(b,d),(c,a),(c,b),(c,d),(d,a),(d,b),(d,c)\}\right\vert =4^{\underline {2}}=4\times 3=12}
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