How Many Functions Have f(1) = f(2)?

[iamsmooth]

Homework Statement

Let A = {1,2,3} and B = {1,2,3,4,5}
Find the number of functions f: A -> B so that f(1) = f(2)

Homework Equations

The Attempt at a Solution

I'm just reviewing random questions for my final on Tuesday and I came upon this question. Seems to be a counting question, and I'm not that sure how to do this since counting questions are more intuition than method (at least that's what I think). Here's what I got:

So a function sends an element from 1 set to another.

There are 3 possible values for the f function, namely f(1), f(2), and f(3).

There are 75 possibilities (5*5*5). So I guess you have to pick the possible values of f(1), f(2) and f(3). So f(1) has 5 choices, f(2) has 1 choice because it must match what f(1) was mapped to, f(3) can be whatever, so it's 5 again.

5 * 1 * 5 = 25

So there are 25 possible functions.

Is this correct? I always hated counting problems ><

 

[hgfalling]

Yes, this is right, except that 5*5*5 = 125, not 75.

This then leads to the intuitive result that 1/5 (25/125) of possible functions have f(1) = f(2).