Divide objects into parts

Question

I want to divide x distinct objects in some specified groups.. Lets say 3 groups of a,b,c number of I am able to find when objects are similar but not in this case.

Please elaborate.Can't understand. – Rusty Jun 16 '13 at 14:24 — Rusty, Jun 16 '13 at 14:24

score 0 · Answer 1 · answered Jun 16 '13 at 14:33

Here $a+b+c=x$

Case 1: $a,b,c$ are pairwise unequal.

Choose $a$ elements for the first group from $x$ elements at first in ${x \choose a}$ ways. Then choose $b$ elements for the second group from the rest elements in ${x-a \choose b}$ ways . Remaining are the $c$ elements for the 3rd group.So the total no. of choices are ${x \choose a}{x-a \choose b}$

case 2: Only two of $a,b,c$ are equal.Let W.L.O.G $a=b$

Choose $a$ elements for the first group from $x$ elements at first in ${x \choose a}$ ways. Then choose $b=a$ elements for the second group from the rest elements in ${x-a \choose a}$ ways . Remaining are the $c$ elements for the 3rd group. So the no. of choices are ${x \choose a}{x-a \choose a}$. But in this case we have to divide this by $2!$(why?) to get the total no. of choices.So total no. of choices are $\frac12{x \choose a}{x-a \choose a}$

case 3: $a=b=c$

Choose $a$ elements for the first group from $x$ elements at first in ${x \choose a}$ ways. Then choose $b=a$ elements for the second group from the rest elements in ${x-a \choose a}$ ways . Remaining are the $c$ elements for the 3rd group. So the no. of choices are ${x \choose a}{x-a \choose a}$. But in this case we have to divide this by $3!$(why?) to get the total no. of choices.So total no. of choices are $\frac1{3!}{x \choose a}{x-a \choose a}$

Divide objects into parts

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