The number of strictly increasing functions f from the set {1, 2, 3, 4, 5, 6} to the set {1, 2, 3, ... , 9} such that $$f(i)\neq i \text{ for }1\leq i\leq 6$$, is equal to:

Total strictly increasing functions:
$$\binom{9}{6}=84$$

Count functions with at least one fixed point (f(i)=i)

If (f(k)=k), then:

So number with (f(k)=k):
$$\binom{9-k}{6-k}$$

Subtract those with at least one (f(i)=i) (via inclusion-exclusion): (56)
84 - 56 = 28

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