Put 25 marbles into 3 piles, each has an odd # of marbles. How many ways can you do this???
Do you have the formula?
Clearly, since 3 does not divide evenly into 25, you cannot do this without some marbles left over.
23-1-1,21-1-3,19-1-5,19-3-3, and so on
To find the number of ways to distribute 25 marbles into 3 piles, each having an odd number of marbles, we can divide the problem into two steps:
Step 1: Determine the maximum number of marbles that can be in each pile.
Since each pile needs to have an odd number of marbles, the maximum number of marbles that can be in each pile is 25 - 2 = 23. This is because if a pile has 23 marbles, at least one marble must be removed to make it an odd number. Similarly, if a pile has 24 marbles, none can be removed to make it an odd number.
Step 2: Determine the number of ways to distribute the marbles.
To find the number of ways to distribute the marbles, we can use a combination formula or a counting technique called "stars and bars."
Stars and Bars method:
Imagine representing the marbles as stars (*) and the divisions between the piles as bars (|). For example, if we have 5 marbles and 2 piles, it can be represented as: **|*** (2 marbles in the first pile, 3 marbles in the second pile).
The number of ways to distribute the marbles is equivalent to the number of ways to arrange these stars and bars. In this case, we have 25 marbles and 3 - 1 = 2 bars.
Using the formula, the number of ways to arrange (25 + 2) objects (25 marbles + 2 bars) with 2 identical bars is given by:
C(n + r - 1, r - 1)
In this case, n = 25 (number of marbles), r = 2 (number of bars). Plugging these values into the formula, we get:
C(25 + 2 - 1, 2 - 1) = C(26, 1) = 26
So, there are 26 different ways to distribute the 25 marbles into 3 piles, each having an odd number of marbles.