Oliver bought 48 chocolate bars from the candy store. He wanted to
pack all the chocolate bars equally into bags. He must have at least
3 bags of chocolate bars and at least 4 chocolate bars in each bag
How many different ways can he pack his chocolate bars?
1) 4
2) 5
3) 6
4) 10
b is the number of bags, c is the number of choc bars
48 = bc , b≥3, c≥4
bc = 3(16)
bc = 4(12)
bc = 6(8)
bc = 8(6)
bc = 12(4) , so I count 5 ways
To find the number of different ways Oliver can pack his chocolate bars, we can use the concept of integer partition.
First, let's represent Oliver's chocolate bars as a sequence of dots, where each dot represents one chocolate bar:
........ ........ ........ ........ ........ ........ ........ ........
Next, let's partition the dots into bags, ensuring that each bag has at least 4 chocolate bars. We can represent the partitions with vertical lines '|'. For example, one possible partition is:
||| || ||| || || |||| ||| ||||
To satisfy the condition of having at least 4 chocolate bars in each bag, we need to place at least three vertical lines before each set of dots. We can represent the minimum number of vertical lines required using the equation:
3 * (number of bags)
Since Oliver must have at least 3 bags, the minimum number of vertical lines required would be:
3 * 3 = 9
Now, let's consider the remaining dots and vertical lines. These represent the additional chocolate bars that can be added to each bag. Oliver must distribute the remaining chocolate bars in a way that ensures each bag has at least 4 chocolate bars.
Since Oliver bought 48 chocolate bars in total, we can represent the remaining dots and vertical lines as a sequence:
........ ........ ........ ........ ........ ........ .
To find the number of different ways to arrange the remaining chocolate bars and vertical lines, we can imagine placing the remaining dots and vertical lines in a line. We can then count the number of ways to arrange the dots, which corresponds to different ways of distributing the remaining chocolate bars.
Using the concept of combination, we can calculate the number of ways to arrange the remaining dots as:
(combination of remaining dots + remaining vertical lines) = (number of remaining dots + number of remaining vertical lines) choose (number of remaining dots)
In this case, the number of remaining dots is 48 - (4 * 3) = 36, and the number of remaining vertical lines is 9. Plugging these values into the combination formula, we get:
(36 + 9) choose (36) = 45 choose 36.
Using the formula for combination, we can calculate:
45 choose 36 = 45! / (36! * (45-36)!) = 45! / (36! * 9!)
To calculate this value, we can use a calculator or spreadsheet software. The result is a large number.
Therefore, the correct answer is not one of the given options (1, 2, 3, or 4). The number of different ways Oliver can pack his chocolate bars exceeds the options provided.
To find the number of different ways Oliver can pack his chocolate bars, we need to consider the restrictions given: he must have at least 3 bags and at least 4 chocolate bars in each bag.
Let's start by finding the maximum number of chocolate bars that can be in each bag. Since Oliver bought 48 chocolate bars and each bag must have at least 4 chocolate bars, the maximum number of chocolate bars in each bag is 48/4 = 12.
Now, we can consider the possible numbers of chocolate bars in each bag. The numbers must be factors of 48 and be between 4 and 12 (inclusive).
The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
Out of these factors, only 4, 6, 8, and 12 are between 4 and 12.
Therefore, Oliver can pack his chocolate bars in 4 different ways: each bag containing either 4, 6, 8, or 12 chocolate bars.
So the correct answer is 1) 4.