Question

This problem concerns functions \(f:\\{1,2,3,4,5,6,7,8\\} \rightarrow\\{0,1,2,3,4,5,6\\} .\) How many such functions have the property that \(\left|f^{-1}(\\{2\\})\right|=4 ?\)

Short answer

The total number of such functions is \(\binom{8}{4} * 7^4\) .

Step-by-step solution

Understand the size of inverse image

First of all, note that \(|f^{-1}(\{2\})|=4 \), means that the value 2 is mapped by exactly 4 numbers from the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\).

Count the ways to choose 4 numbers from domain set

There are \(\binom{8}{4}\) ways to choose 4 elements from the domain set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\) that map to 2. Here, \(\binom{8}{4}\) stands for the binomial coefficient or '8 choose 4', which gives the number of ways to choose 4 elements from a set of 8 elements without considering the order.

Count the ways to assign remaining numbers

Now, for the remaining 4 elements in the domain, each can map to any of the 7 elements in the codomain. Therefore, there are \(7^4\) ways to assign these mappings.

Find the total number of functions

Since the choices in Step 2 and Step 3 are independent, by the multiplication principle of counting, the total number of such functions is given by \(\binom{8}{4} * 7^4\).