# ordered and unordered partitions

• Oct 12th 2009, 04:11 AM
novice
ordered and unordered partitions
In how many ways can 10 students be divided into three teams, one containing 4 students and the others 3?

Method 1 makes sense, in which the answer is $\displaystyle \frac{10!}{4!3!3!}\cdot\frac{1}{2}$.

But method 2 bothers me, where the answer is $\displaystyle \binom{10}{4}\binom{5}{2}$.

I can understand choosing 4 out of 10 the first time, but what is the logical explanation for choosing 2 out 5 the second time? This does not seem very consistent.
• Oct 12th 2009, 10:51 AM
Opalg
Quote:

Originally Posted by novice
In how many ways can 10 students be divided into three teams, one containing 4 students and the others 3?

Method 1 makes sense, in which the answer is $\displaystyle \frac{10!}{4!3!3!}\cdot\frac{1}{2}$.

But method 2 bothers me, where the answer is $\displaystyle \binom{10}{4}\binom{5}{2}$.

I can understand choosing 4 out of 10 the first time, but what is the logical explanation for choosing 2 out 5 the second time? This does not seem very consistent.

Imagine you're the coach for this squad. You pick four of the ten students for the first team ($\displaystyle \textstyle{10\choose4}$ ways of doing that). You then designate one of the other six students (doesn't matter who it is) to be a team captain, with instructions to choose two of the remaining five students to complete his/her team ($\displaystyle \textstyle{5\choose2}$ ways of doing that). The three left-over students form the third team, of course.
• Oct 12th 2009, 11:39 AM
novice
Quote:

Originally Posted by Opalg
Imagine you're the coach for this squad. You pick four of the ten students for the first team ($\displaystyle \textstyle{10\choose4}$ ways of doing that). You then designate one of the other six students (doesn't matter who it is) to be a team captain, with instructions to choose two of the remaining five students to complete his/her team ($\displaystyle \textstyle{5\choose2}$ ways of doing that). The three left-over students form the third team, of course.

Thank you, Opalg, for taking time answering my question. Your explanation makes a lot of sense. We need more of your kind to write math books.