Combination Math Problem: Choosing 5 Balls from 4 Boxes without Restriction

In summary, there are 1024 ways to choose a set of five balls without restriction from four boxes containing different colored balls. This can be calculated by taking 4 to the power of 5, since there are four colors and five balls in the set. When considering combinations, one must also consider cases where all colors are the same or where there are combinations of two same colors and three different colors.
  • #1
Harmony
203
0
Question Statement
4 boxes each contain a large number of identical balls, those in 1 box are red, those in the 2nd box are blue, those in the 3rd box are yellow and those in the remaining box are green. In how many ways can a set of five balls be chosen? (without restriction)

My thoughts
If the question is asking for the permutation, then the answer is 4^5 (since there is 4 colour and the colour can be repeated)

But the question is asking for combinations. How should I attempt this question? Should I consider cases where all colours are the same/ different, 2 same 3 different...etc.?
 
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  • #2
Harmony said:
But the question is asking for combinations. How should I attempt this question? Should I consider cases where all colours are the same/ different, 2 same 3 different...etc.?
yes...you can have all of same color , or two of same color and 3 of 3 different colors ans so on
 
  • #3


I would approach this problem by first defining the variables and parameters involved. In this case, we have 4 boxes containing a large number of identical balls, with each box representing a different color (red, blue, yellow, and green). We are asked to choose 5 balls from these 4 boxes without any restrictions.

To solve this problem, we can use the combination formula, which is nCr = n! / r!(n-r)!, where n is the total number of items and r is the number of items we want to choose. In this case, n = 4 and r = 5.

However, since we are not restricted to choosing only one ball from each box, we need to consider different cases. For example, we can choose all 5 balls from the same box, or we can choose 3 balls from one box and 2 balls from another box, etc.

To account for all these cases, we can break down the problem into smaller sub-problems and add them together. For instance, we can calculate the number of ways to choose 5 balls from one box, and then multiply it by 4 (since there are 4 boxes in total). This would give us the total number of combinations if all 5 balls were of the same color.

Similarly, we can calculate the number of ways to choose 3 balls from one box and 2 balls from another box, and then multiply it by the number of ways to choose 2 different boxes from the remaining 3 boxes. This would give us the total number of combinations if we have 3 balls of one color and 2 balls of another color.

We can continue this process for all the possible combinations and add them together to get the final answer. Alternatively, we can use the combination formula and substitute n = 4 and r = 5, but this would require us to consider all the possible combinations in a systematic manner.

In conclusion, the number of ways to choose 5 balls from 4 boxes without any restrictions would be the sum of all the possible combinations, taking into account the different cases. This approach allows us to solve the problem systematically and efficiently.
 

1. What is a combination math problem?

A combination math problem is a type of mathematical problem that involves selecting a group of objects from a larger set, without taking into account the order in which the objects are selected.

2. How do you solve combination math problems?

To solve a combination math problem, you can use the formula nCr = n! / r!(n-r)!, where n represents the total number of objects and r represents the number of objects being selected. You can also use a combination calculator or list out all possible combinations to find the answer.

3. What is the difference between a combination and a permutation?

The main difference between a combination and a permutation is that a combination does not take into account the order in which the objects are selected, while a permutation does. In a combination, selecting the same objects in a different order is considered the same combination, while in a permutation, it is considered a different permutation.

4. Can combination math problems be applied in real life?

Yes, combination math problems can be applied in many real-life situations. For example, when selecting a team of players from a larger pool, or choosing which items to buy from a set menu, or creating a password from a set of numbers and letters.

5. Are there any shortcuts or tricks to solve combination math problems?

There are a few shortcuts or tricks that can help you solve combination math problems faster. One example is the use of the Pascal's Triangle to find the number of combinations for a given set of objects. Additionally, familiarizing yourself with the combination formula and practicing with various examples can also improve your speed and accuracy in solving these types of problems.

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