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Chapter 1: Problem 17

How many ways are there to place 12 marbles of the same size in five distinct jars if (a) the marbles are all black? (b) each marble is a different color?

Short Answer

Expert verified
For case (a) where all the marbles are the same color, there are 1820 ways of arranging them in the jars. For case (b) where each marble is a different color, there are 244140625 ways.

Step by step solution

01

Case (a) Solution

In case (a) the marbles are indistinguishable, making this a typical problem of combinations with repetitions allowed. Combinations with repetitions are usually solved by the stars and bars method. With this method, you consider that there are 12 marbles ('stars') and 4 separators ('bars') to partition them into 5 jars. So the total number of ways is: \( C(n + k - 1, k - 1) = C(12 + 5 - 1, 5 - 1) = C(16, 4) \).
02

Calculation of Combinations

In order to calculate combinations we use the formula: \( C(n, k) = n! / [k!(n-k)!] \), where '!' stands for factorial, which means the product of an integer and all the integers below it. Now we can substitute the values into the formula: \( C(16, 4) = 16! / [4!(12)!] \).
03

Calculation of Factorials

Now substitute the factorials: \( 16! = 16*15*14*13*12!, 4! = 4*3*2 \). After simplifying the expression, the result is: \( 1820 \). So, there are 1820 ways to arrange 12 indistinguishable marbles into 5 jars.
04

Case (b) Solution

If each marble is a different color (marbles are distinguishable), this becomes a problem of distributing distinct objects into distinct groups. The solution is given by: \( k^n \), where 'k' is the number of jars, and 'n' is the number of marbles. So we have: \( 5^{12} \). The total number of ways the marbles can be distributed, can be calculated.
05

Calculation of Expression

Calculate the expression \( 5^{12} \). The result is \( 244140625 \). Therefore, there are \( 244140625 \) ways to arrange 12 distinguishable marbles into 5 jars.

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