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# In how many ways can 4 people be seated in a row of 12 chairs?

I broke the question down into 2 parts, then solved them and multiplied the answer to get 11,880 ways.

We can solve this by breaking down the question into 2 smaller questions. The first question is "how many ways can we choose 4 chairs out of 12 to have someone sit in" and the second question is "how many ways can 4 people sit in 4 chairs". Our final answer will be the product of the 2 smaller questions.

The first smaller question - "how many ways can we choose 4 chairs out of 12 to have someone sit in" - is a combinations question (how many ways can we pick 4 chairs from a group of 12) - that is written in a few different notations - I'll write it out as Combination 12 choose 4 and in notation ##C_(12,4)##

For each combination of 4 chairs selected out of the 12, there are different ways the people can sit in them - we can have persons A B C D, or A B D C, etc. There are, in fact ##4!## ways each selection of chairs can be sat in.

Our final answer, therefore is

##(4!)(C_(12,4))##

Let's solve

##(4!)=4*3*2*1=24##

##(C_(12,4))=(12!)/((4!)(8!))=(12*11*10*9*8!)/((24)(8!))=(12*11*10*9)/24##

Now let's combine the 2:

##(24)((12*11*10*9)/24)=(12*11*10*9)=11,880##