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The number of ways of distributing then identical fruits amongst four boys such that any boy may secure any number of roots and not all fruits need to be distributed is?
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The number of ways of distributing then identical fruits amongst four ...
Understanding the Problem
To distribute ten identical fruits among four boys, we need to consider that each boy can receive any number of fruits, and it’s not mandatory for all fruits to be distributed.
Applying the Stars and Bars Theorem
This problem can be solved using the Stars and Bars theorem, which is a popular combinatorial method used for distributing indistinguishable objects into distinguishable boxes.
Steps to Solve
- Total Fruits: 10 (identical)
- Total Boys: 4 (distinguishable)
We need to find the number of non-negative integer solutions to the equation:
- x1 + x2 + x3 + x4 ≤ 10
where x1, x2, x3, and x4 represent the number of fruits received by each boy.
Transforming the Equation
To convert the inequality into an equation, we introduce a variable:
- Let x5 be the number of fruits not distributed.
Now, we rewrite our equation:
- x1 + x2 + x3 + x4 + x5 = 10
Calculating the Solutions
According to the Stars and Bars theorem, the number of non-negative integer solutions to the equation is given by:
- (n + k - 1) choose (k - 1)
where n is the total number of fruits (10) and k is the number of boys (4).
So, we calculate:
- (10 + 4 - 1) choose (4 - 1) = 13 choose 3
Final Calculation
Calculating 13 choose 3 gives us:
- 13! / (3! * (13 - 3)!) = 286
Thus, the total number of ways to distribute the ten identical fruits among four boys is 286.
To distribute ten identical fruits among four boys, we need to consider that each boy can receive any number of fruits, and it’s not mandatory for all fruits to be distributed.
Applying the Stars and Bars Theorem
This problem can be solved using the Stars and Bars theorem, which is a popular combinatorial method used for distributing indistinguishable objects into distinguishable boxes.
Steps to Solve
- Total Fruits: 10 (identical)
- Total Boys: 4 (distinguishable)
We need to find the number of non-negative integer solutions to the equation:
- x1 + x2 + x3 + x4 ≤ 10
where x1, x2, x3, and x4 represent the number of fruits received by each boy.
Transforming the Equation
To convert the inequality into an equation, we introduce a variable:
- Let x5 be the number of fruits not distributed.
Now, we rewrite our equation:
- x1 + x2 + x3 + x4 + x5 = 10
Calculating the Solutions
According to the Stars and Bars theorem, the number of non-negative integer solutions to the equation is given by:
- (n + k - 1) choose (k - 1)
where n is the total number of fruits (10) and k is the number of boys (4).
So, we calculate:
- (10 + 4 - 1) choose (4 - 1) = 13 choose 3
Final Calculation
Calculating 13 choose 3 gives us:
- 13! / (3! * (13 - 3)!) = 286
Thus, the total number of ways to distribute the ten identical fruits among four boys is 286.
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Question Description
The number of ways of distributing then identical fruits amongst four boys such that any boy may secure any number of roots and not all fruits need to be distributed is? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The number of ways of distributing then identical fruits amongst four boys such that any boy may secure any number of roots and not all fruits need to be distributed is? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of ways of distributing then identical fruits amongst four boys such that any boy may secure any number of roots and not all fruits need to be distributed is?.
The number of ways of distributing then identical fruits amongst four boys such that any boy may secure any number of roots and not all fruits need to be distributed is? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The number of ways of distributing then identical fruits amongst four boys such that any boy may secure any number of roots and not all fruits need to be distributed is? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The number of ways of distributing then identical fruits amongst four boys such that any boy may secure any number of roots and not all fruits need to be distributed is?.
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