SOLUTION: Smith Welding is setting up the shop for a new production run. Nine welding machines are available but only three spaces are available in the production area of the shop. In how ma

Question 1084056: Smith Welding is setting up the shop for a new production run. Nine welding machines are available but only three spaces are available in the production area of the shop. In how many different ways can the nine welding machines be arranged in the three available spaces? (permutation)
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Answer: 504
With permutations, order does matter.
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Explanation:

There are two methods we can use to get the answer.

Method 1) This method, in my opinion, is the more intuitive and easier to use method.

We have 3 slots (or spaces) to fill.
Let's call them slot A, slot B, slot C

Slot A has 9 choices
Slot B has 8 choices
Slot C has 7 choices
Keep in mind that once you make a choice, you can't reuse it
This is why we have a countdown.

Multiply out those values: 9*8*7 = 504 which is the answer.

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Method 2) For this method, we'll use the nPr formula

Plug n = 9 and r = 3 into the nPr permutation formula to get
nPr = (n!)/((n-r)!)
9P3 = (9!)/((9-3)!)
9P3 = (9!)/(6!)
9P3 = (9*8*7*6!)/(6!)
9P3 = 9*8*7 ** Note **
9P3 = 504
which yields the same result as shown in method 1.

The line marked with ** Note ** shows the expression 9*8*7 which is what we did for method 1.
This expression is the result of the terms "6!" canceling out.

So in a sense, method 1 is buried in the much longer method 2.

In this case, method 1 is better to use since there a small number of slots to fill. As more slots are needed, I recommend using the permutation formula. Of course, if your teacher requires you to use the permutation formula then listen to your teacher.

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