Question

How many ways are there to distribute 12 distinguishable objects into six distinguishable boxes so that two objects are placed in each box?

Answer 1

hello friend

Answer 2

here we must use the fundamental principle of counting...

Answer 3

the first box can actually fill in 12 possible ways...and then the second box can be filled in 11 ways only...

Answer 4

totally there arre 12*11 ways to fill these boxes...
therefore answer=132

Answer 5

The number of combinations of the 12 objects taken 2 at a time is found as follows:
There are 12 choices for the first object and 11 choices for the second object. Therefore the number of possible pairs is (12 * 11)/2 = 66. Note that we divide by 2, the reason being that the order of choice does not matter.
The number of combinations of the 66 pairs taken 6 at a time is given by:
\[\large 66C6=\frac{66\times65\times64\times63\times62\times61}{6\times5\times4\times3\times2\times1}=you\ can\ calculate\]

Answer 6

Unfortunately, the answer is 7,484,400

Answer 7

This is how to do it if you wanted to know!
http://math.stackexchange.com/questions/468824/distinguishable-objects-into-distinguishable-boxes
In this case, \[\left(\begin{matrix}12 \\ 2\end{matrix}\right) \left(\begin{matrix}10 \\2\end{matrix}\right)\left(\begin{matrix}8 \\ 2\end{matrix}\right)\left(\begin{matrix}6 \\ 2\end{matrix}\right)\left(\begin{matrix}4 \\ 2\end{matrix}\right)\left(\begin{matrix}2 \\ 2\end{matrix}\right)\]