TITLE: If the operations can be done in any order, how many different ways of completing the manufacturing are possible?
QUESTION [0 upvotes]: Question: In a sheet metal operation, three notches and four
bends are required. If the operations can be done in any order,
how many different ways of completing the manufacturing are
possible?
While the answer solution says 35, I am having trouble understanding how to apply to combination formula to this question. I know there are 7 operations total which makes n = 7. However, the selection size is also 7, which shouldn't also make k = 7 as well?

REPLY [0 votes]: I'm not sure what your $n$ and $k$ are, but we can think of the problem in this way. Let $N$ denote a notch and $B$ denote a bend. How many ways can we put 3 $N$'s and 4 $B$'s in $7$ slots?
$$
\text{__} \; \text{__} \; \text{__} \; \text{__} \; \text{__} \; \text{__} \; \text{__} \;
$$
Here is one way:
$$
\text{N} \; \text{N} \; \text{N} \; \text{B} \; \text{B} \; \text{B} \; \text{B} \;
$$
In order to count the total number of ways, think of how many ways we can place the first $N$, then the second $N$, and then the third $N$. Do we need to account for the $B$'s afterwards? Does this remind you of any formulas you might have learned?
Now this more abstract formulation is essentially the same as your problem; each sequence of $N$'s and $B$'s gives us a different manufacturing order.
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