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6!/(6-3)! is the formula we would use if there were no blanks or spaces.

But that just gives us the order of the people, and doesn't account for spaces between them. People could be arranged:
ABCXXX XABCXX XXABCX XXXABC
ABXCXX XABXCX XXABXC
ABXXCX XABXXC
ABXXXC

AXBCXX XAXBCX XXAXBC
AXBXCX XAXBXC
AXBXXC

AXXBCX XAXXBC
AXXBXC

AXXXBC

20 different ways

The solution is 20*6!/(6-3)!

But I don't know the formula, so I can't answer the question.

Can we think of this problem as similar to finding the possible anagrams of the word ABCXXX? A-ha!

Yes and no. If you did that, you would have all possible permutations of ABC and the three spaces, but what about C, D, and E?

Maybe do this in two steps:
1. Number of COMBINATIONS from 6c3, which is 20; multiply by
2. Number of anagrams from ABCXXX, which is 6!/(3!*1!*1!*1!), which is 120.

Total=2400.

Math is the same as my previous solution but this seems more sensible.

And the real killer is that a reasonably dedicated high school senior could this in his sleep...

Can we think of this problem as similar to finding the possible anagrams of the word ABCXXX? A-ha!

Yes and no. If you did that, you would have all possible permutations of ABC and the three spaces, but what about C, D, and E?

Maybe do this in two steps: 1. Number of COMBINATIONS from 6c3, which is 20; multiply by 2. Number of anagrams from ABCXXX, which is 6!/(3!*1!*1!*1!), which is 120.

Total=2400.

Math is the same as my previous solution but this seems more sensible.

And the real killer is that a reasonably dedicated high school senior could this in his sleep...

hey stoolfi,

we can pick 3 chairs out of 6 in 20 ways...

and then *each* of these "Ways"...seat these three..

So these three can be seated in 3! ways..isnt it.

And we have 20 of these combinations.

so...the total for 20 combination is 20 * 3! = 120 ways

Wow...I'm still reeling from my HBS admit . Thank you once again to everyone who has helped me through this process. Every year, USNews releases their rankings of...

Almost half of MBA is finally coming to an end. I still have the intensive Capstone remaining which started this week, but things have been ok so far...