With 50 pennies in three jars labeled A, B and C, how may ways can you put the pennies in the jars assuming they are identical with at least two pennies in each jar?
From the 50 pennies, put 2 in each of the jars to satisfy the constraint.
Now we need to distribute the remaining 46 into 3 jars.
Use generating function method to find the coefficient of a^44.
See http://www.jiskha.com/display.cgi?id=1319753112
for an example of the generating function.
Sorry, there are 44 remaining pennies.
I got 1035 using the generating function.
To determine the number of ways to distribute 50 pennies among three jars labeled A, B, and C, where each jar must have at least two pennies, we can follow these steps:
Step 1: Assign two pennies to each jar
Since each jar must have at least two pennies, let's start by distributing two pennies to each jar. This leaves us with (50 - 6) = 44 pennies to distribute further.
Step 2: Distribute the remaining pennies among the jars
We can distribute the remaining 44 pennies among the three jars in various ways to find the total number of arrangements. To do this, we can use a concept called "stars and bars" or "balls and urns."
We have 44 pennies to distribute among three jars, which can be represented as placing 44 indistinguishable stars (pennies) into three distinct jars, represented as partitions (|) or bars.
For example, if we have 44 pennies and 3 jars, we could represent it as:
**|***|************ <-- 44 pennies and 3 jars
To find the number of ways to distribute these pennies, we need to count the number of unique arrangements of stars and bars.
Step 3: Calculate the arrangements using combinations
The number of unique arrangements can be calculated using combinations. We need to choose two positions out of the 44+3-1 = 46 available positions for bars. This is because we have 44 pennies to distribute between the jars and 3 jars to place them into.
Using the combination formula, we can calculate it as C(46,2) = 46! / (2!(46-2)!) = 46! / (2!44!) = (46*45) / 2 = 1035
Therefore, there are 1035 different ways to distribute 50 pennies among three jars labeled A, B, and C, with at least two pennies in each jar.
To find the number of ways you can distribute the 50 pennies among three jars labeled A, B, and C, we can use a technique called "stars and bars" or "balls and urns."
Since each jar must have at least two pennies, let's first distribute 2 pennies to each jar. This leaves us with 50 - 2 × 3 = 44 pennies that need to be distributed among the three jars.
To solve this, we can imagine a scenario where we have 44 pennies (stars) and two walls (bars) to separate the pennies into three groups (jars). Thus, we have 44 + 3 - 1 = 46 positions (stars and bars).
Now, we need to choose 2 out of the 46 positions to place the bars. The other positions will automatically be filled with stars (pennies). This is equivalent to choosing the positions for the bars out of the total positions available.
Using the combination formula, we can calculate the number of ways to choose 2 positions out of 46:
C(46, 2) = 46! / (2! × (46 - 2)!) = 46! / (2! × 44!) = (46 × 45) / 2 = 1,035
Therefore, there are 1,035 ways to distribute the 50 pennies among the three jars labeled A, B, and C, assuming they are identical and each jar has at least two pennies.