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Combinations Question

In how many ways can 12 people be divided into:
(a) two equal groups
(b) three equal groups

(a) Using 12C6 makes sense but the answer requires that value to be divided by 2 for some reason that I don't understand.

(b) 12C4*8C4*4C4 again makes sense, but why in this case is the result divided by 3!
Original post by xlaser31
In how many ways can 12 people be divided into:
(a) two equal groups
(b) three equal groups

(a) Using 12C6 makes sense but the answer requires that value to be divided by 2 for some reason that I don't understand.

(b) 12C4*8C4*4C4 again makes sense, but why in this case is the result divided by 3!

Division is needed due to overcounting.

Imagine there are four people; A, B, C, D.

The following are the only possibilities for two equally sized groups:

AB CD
AC BD
AD BC

Calculating 4C2 = 6 involves those three pairings BUT ALSO the following additional pairings:

BC AD
BD AC
CD AB

However, these latter three groupings are equivalent to the first three. So actually, 4C2 overcounts the groupings by factor of 2!=2.

Similarly with three groups you would overcount by factor of 3!=6.
(edited 4 months ago)

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