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!

(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!

(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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