Thread: Counting problem

1. Counting problem

Hi,
I have a problem with this exercise - How many ways are there of putting 15 pennies into three piles (each with at least one penny) such that each pile contains an odd number of pennies? I think that the answer is seven, but have difficulties to write it down in a formal way. I would be greateful for any help.
Thanks in advance...

2. If the ‘piles’ are unordered then the answer is clearly seven. A simple listing will show that. I do think that is what the problem means.

However, if the ‘piles’ are ordered then the answer is twenty-eight. I don’t think that is what is meant.

3. Hello, Gibo!

I agree completely with Plato's explanation.

$\displaystyle \text{How many ways are there of putting 15 pennies into 3 piles}$
$\displaystyle \text{(each with at least one penny) such that each pile contains}$
$\displaystyle \text{an odd number of pennies?}$

I tried to find a more formal approach to this problem,
. . but ended up Listing the cases anyway.

Place one penny in each pile: .$\displaystyle |\;\circ\;|\;\circ\;|\;\circ \;|$

Form the other 12 pennies into pairs:. $\displaystyle (\circ\:\circ)\;(\circ\:\circ)\;(\circ\:\circ) \;(\circ\:\circ)\;(\circ\:\circ)\;(\circ\:\circ)$

Now distribute the pairs among the three piles. .There are seven ways:

. . $\displaystyle [0,0,6],\;[0,1,5],\;[0,2,4],\;[0,3,3],\;[1,1,4],\;[1,2,3],\;[2,2,2]$

4. Thanks a lot for help