Number of functions from a to b where {123} is in the range of (f)

[Dank2]

Homework Statement

A has n elements.
B={0,1,2,3}
{1,2,3}⊆range(f)

Homework Equations

The Attempt at a Solution

So in each function we must choose those 3 numbers in the range.
So let's first choose all the diffrent possiblites to choose those 3:
n*(n-1)*(n-2)

now for the remaining elemnts, we should have 4^n-3

So totally we should have n*(n-1)*(n-2) * 4^n-3 diffrent functions.

 

[mfb]

You can't separate them like that. You would e.g. count (1,2,3,3) twice, once with the first 3 "chosen" and once with the second one.
You can use inclusion-exclusion: How many functions are there in total? How many without 3? How many without 2? Without 1? How many did you subtract twice now?