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In how many ways can 5 different balls be distributed to 4 different boxes, when each of the can hold any no of ball ?
- a)1024
- b)1200
- c)1234
- d)1600
- e)None of these
Correct answer is option 'A'. Can you explain this answer?
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In how many ways can 5 different balls be distributed to 4 different b...
Answer – A.1024 Explanation : No of way =45 = 1024
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In how many ways can 5 different balls be distributed to 4 different b...
Solution:
The given problem can be solved using the stars and bars method.
Let's consider the 5 balls as stars and the 4 boxes as bars, such that the balls are distributed among the boxes as per their position with respect to the bars.
For example, let's say we have 5 balls and 4 boxes, then the distribution can be represented as:
* | * * | * * | *
This means that the first box has 1 ball, the second box has 2 balls, the third box has 2 balls, and the fourth box has 0 balls.
Using this method, we can determine the number of ways in which the balls can be distributed among the boxes.
The number of ways can be calculated using the formula:
Number of ways = (n + k - 1)C(k - 1)
Where n is the number of balls and k is the number of boxes.
Substituting the values, we get:
Number of ways = (5 + 4 - 1)C(4 - 1)
= 8C3
= 56
Therefore, the number of ways in which 5 different balls can be distributed to 4 different boxes is 56.
But in the options, the answer is given as 1024. This is because the question is asking for the number of ways in which each box can hold any number of balls. So, each ball can be placed in any of the 4 boxes, giving us 4 choices for each ball. Therefore, the total number of ways becomes:
Number of ways = 4^5
= 1024
Hence, the correct answer is option A) 1024.
The given problem can be solved using the stars and bars method.
Let's consider the 5 balls as stars and the 4 boxes as bars, such that the balls are distributed among the boxes as per their position with respect to the bars.
For example, let's say we have 5 balls and 4 boxes, then the distribution can be represented as:
* | * * | * * | *
This means that the first box has 1 ball, the second box has 2 balls, the third box has 2 balls, and the fourth box has 0 balls.
Using this method, we can determine the number of ways in which the balls can be distributed among the boxes.
The number of ways can be calculated using the formula:
Number of ways = (n + k - 1)C(k - 1)
Where n is the number of balls and k is the number of boxes.
Substituting the values, we get:
Number of ways = (5 + 4 - 1)C(4 - 1)
= 8C3
= 56
Therefore, the number of ways in which 5 different balls can be distributed to 4 different boxes is 56.
But in the options, the answer is given as 1024. This is because the question is asking for the number of ways in which each box can hold any number of balls. So, each ball can be placed in any of the 4 boxes, giving us 4 choices for each ball. Therefore, the total number of ways becomes:
Number of ways = 4^5
= 1024
Hence, the correct answer is option A) 1024.
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