mathmath333 one year ago Counting question

1. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{The number of ways of dividing 10 different balloons }\hspace{.33em}\\~\\ & \normalsize \text{in two groups each containing 5 balloons is ? }\hspace{.33em}\\~\\ \end{align}}

2. Michele_Laino

I think that the requested number is: $\Large 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 = \left( {\begin{array}{*{20}{c}} {10} \\ 5 \end{array}} \right) \cdot 5!$

3. mathmath333

but answer given is $$\large \dfrac{10!}{5!5!2!}$$

4. Michele_Laino

I'm sorry!

5. mathmath333

sometimes their is typo in book too

6. Michele_Laino

I don't think that it is a typo, I think that my answer is incorrect!

7. Michele_Laino

if we have n elements, then for each subset of k elements I have a subset of n-k elements, we have: $\Large \left( {\begin{array}{*{20}{c}} n \\ k \end{array}} \right) = \left( {\begin{array}{*{20}{c}} n \\ {n - k} \end{array}} \right)$

8. mathmath333

this applies for n different elements right

9. Michele_Laino

yes! In our case we have n=10 and k=5

10. mathmath333

ohk thanks

11. Michele_Laino

I think that the requested number of ways is given by the total number of subset divided by 2=2!

12. Michele_Laino

sinceeach way, gives two subsets of 5 elements

13. Michele_Laino

since each*

14. mathmath333

ok