how many ways are there to distribute
a)6 indistinguishable marbles into 6 distinguishable bowls
b)12 distinguishable marbles into 6 distinguishable bowls
c)12 distinguishable marbles into 6 disinguishable bowls so that two marbels are placed in each bowl
(1) 1
(b) Must each bowl have at least one marble?
(c) 12C2 * 6!
I think my answer to (c) is bogus. Still thinking on it.
a) If we have 6 indistinguishable marbles and 6 distinguishable bowls, we can use the concept of "Stars and Bars" to count the number of ways to distribute the marbles.
Using this concept, we can represent each marble as a star and use bars to separate the stars into different bowls. The number of bars we need is one less than the number of bowls.
So, for 6 indistinguishable marbles and 6 distinguishable bowls, we need 5 bars. We can arrange the 6 marbles and 5 bars in a line, resulting in a total of 11 objects.
The number of ways to distribute the marbles would then be the number of ways to choose the positions of the 5 bars out of the 11 objects, which can be calculated using combinations.
Using the formula for combinations, the number of ways to distribute the marbles would be C(11, 5), which is equal to 462.
Therefore, there are 462 ways to distribute 6 indistinguishable marbles into 6 distinguishable bowls.
b) If we have 12 distinguishable marbles and 6 distinguishable bowls, we can think of this as assigning each marble to one of the 6 bowls.
Since each marble can be assigned to any of the 6 bowls, we have 6 options for each of the 12 marbles.
The total number of ways to distribute the marbles would then be 6^12, which is approximately 2,176,782,336.
Therefore, there are approximately 2,176,782,336 ways to distribute 12 distinguishable marbles into 6 distinguishable bowls.
c) If we must place 2 marbles in each bowl, we can consider this as choosing 2 marbles out of the 12 for the first bowl, choosing 2 marbles out of the remaining 10 for the second bowl, and so on.
Using combinations again, we can calculate the number of ways to choose 2 marbles from a set of 12 as C(12, 2), which is equal to 66.
Since there are 6 bowls, we need to calculate the number of ways to distribute the marbles among the 6 bowls, which would be 66^6.
Therefore, there are 66^6 ways to distribute 12 distinguishable marbles into 6 distinguishable bowls so that two marbles are placed in each bowl.
a) To find the ways to distribute 6 indistinguishable marbles into 6 distinguishable bowls, we can use the concept of stars and bars. In this case, each marble represents a star and the bowls can be represented by bars. We need to distribute the stars (marbles) among the bars (bowls) such that each bowl can have any number of marbles (including zero), and each marble is accounted for.
Using stars and bars, we can place 5 bars among the 6 bowls. The distribution can be represented by the following arrangement:
* | * | * | * | * | *
This arrangement indicates the distribution of marbles into bowls. For example, the first bowl has no marbles (to the left of the first bar), the second bowl has one marble (between the first and second bars), the third bowl has one marble (between the second and third bars), and so on.
The number of ways to arrange the stars and bars is (6 + 5) choose 5, which is equivalent to 11 choose 5. Using the formula for combinations, this simplifies to 462. Therefore, there are 462 ways to distribute 6 indistinguishable marbles into 6 distinguishable bowls.
b) To find the ways to distribute 12 distinguishable marbles into 6 distinguishable bowls, each bowl can have any number of marbles (including zero). This is equivalent to finding the number of partitions of 12 into at most 6 parts. We can approach this problem using generating functions.
The generating function for the number of partitions of 12 is given by:
(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)^6
Expanding this expression, we will have terms with different powers of x that correspond to each possible distribution of the marbles. To find the number of ways to distribute the marbles, we need to find the coefficient of x^12 in the expansion.
By performing the expansion and simplification, we get the coefficient of x^12 as 317,520. Therefore, there are 317,520 ways to distribute 12 distinguishable marbles into 6 distinguishable bowls.
c) To find the number of ways to distribute 12 distinguishable marbles into 6 distinguishable bowls such that two marbles are placed in each bowl, we can divide the problem into two steps:
Step 1: Selecting pairs of marbles - Since there are 12 marbles, we need to select 6 pairs of marbles to distribute them equally into the bowls. The number of ways to select pairs from 12 marbles is given by 12 choose 2, 6 times, which is equivalent to (12 choose 2)^6.
Step 2: Distributing the pairs into bowls - Once we have the 6 pairs, we need to distribute them into the 6 distinguishable bowls. Since the pairs are indistinguishable, this becomes equivalent to distributing the 6 indistinguishable marbles into the 6 distinguishable bowls, which we solved in part (a).
Therefore, the total number of ways to distribute 12 distinguishable marbles into 6 distinguishable bowls with two marbles in each bowl is (12 choose 2)^6 * 462, which simplifies to 17,553,600.