A lawn care company has five employees, and there are ten

Nice question!

Take the task of assigning two houses to each employee and break it into stages.

Stage 1: Select 2 houses for employee #1
Since the order in which we select the two houses does not matter, we can use combinations.
We can select 2 houses from 10 houses 10C2 ways [= (10)(9)/(2) ways]
So, we can complete stage 1 in (10)(9)/(2) ways

NOTE: Given the answer choices, I'm not going to evaluate (10)(9)/(2). You'll see why shortly.

Stage 2: Select 2 houses for employee #2
There are 8 houses left to choose from.
So we can complete stage 2 in 8C2 ways (= (8)(7)/(2) ways)

Stage 3: Select 2 houses for employee #3
There are 6 houses left to choose from.
We can complete stage 3 in 6C2 ways (= (6)(5)/(2) ways)

Stage 4: Select 2 houses for employee #4
We can complete stage 4 in 4C2 ways (= (4)(3)/(2) ways)

Stage 5: Select 2 houses for employee #5
We can complete stage 5 in 2C2 ways (= (2)(1)/(2) ways)

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus assign all 10 houses to 5 employees) in [(10)(9)/(2)][(8)(7)/(2)][(6)(5)/(2)][(4)(3)/(2)][(2)(1)/(2)] ways
Evaluate to get: 10!/(2^5)

Answer: . . . hmmm, not among the answer choices.

I Googled the question and found out that it's a question from GMAT For Dummies. They say the answer is D, but give no rationale, other than to plug 10 and 5 into the permutations formula. However, this calculation would be for a scenario in which each worker works at ONLY ONE house (not 2, as in the question)


Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

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