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In how many ways can 25 identical chocolates be given to 4 brothers such that the eldest brother is given at least 4 chocolates, the one just younger to him is given at least 3 chocolates, the second youngest is given at least 2 and the youngest one is given at least 1 chocolate?
- a)28C3
- b)25C3
- c)14C3
- d)15C3
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
In how many ways can 25 identical chocolates be given to 4 brothers su...
Let the youngest brother be given A chocolate, the one elder to him be given B chocolates, the third oldest be given C chocolates and the eldest one be given D chocolates.
Let A= 1+a, so that a≥ 0
B = 2+b, so that b≥ 0
C = 3+c, so that c≥ 0
and D = 4+d, so that d≥ 0.
So, A+B+C+D= 25
⇒ (1+a)+(2+b)+(3+c)+(4+d)=25
⇒ a+b+c+d= 25-10
⇒ a+b+c+d=15, where all of a, b, c and d are ≥ 0
Total number of ways =
Let A= 1+a, so that a≥ 0
B = 2+b, so that b≥ 0
C = 3+c, so that c≥ 0
and D = 4+d, so that d≥ 0.
So, A+B+C+D= 25
⇒ (1+a)+(2+b)+(3+c)+(4+d)=25
⇒ a+b+c+d= 25-10
⇒ a+b+c+d=15, where all of a, b, c and d are ≥ 0
Total number of ways =
(n-1)
C(r-1)
(15-1
)C(4-1)
= 14
C3
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In how many ways can 25 identical chocolates be given to 4 brothers su...
Problem Breakdown
To solve the problem of distributing 25 identical chocolates among 4 brothers with specific minimum requirements, we first need to account for the chocolates that are already allocated based on these requirements.
Minimum Chocolates Distribution
- Eldest brother: At least 4 chocolates
- Second brother: At least 3 chocolates
- Third brother: At least 2 chocolates
- Youngest brother: At least 1 chocolate
Initial Allocation
- Total chocolates given initially = 4 + 3 + 2 + 1 = 10 chocolates
After allocating these minimum required chocolates, we calculate the remaining chocolates:
Remaining Chocolates
- Total chocolates left = 25 - 10 = 15 chocolates
Distribution of Remaining Chocolates
Next, we need to distribute the remaining 15 chocolates among the 4 brothers without any restrictions (since the minimum requirements have already been met).
Applying Stars and Bars Theorem
This is a classic application of the "Stars and Bars" theorem, which states that the number of ways to distribute n identical items into r distinct groups is given by the formula:
- (n + r - 1) choose (r - 1)
Here, n = 15 (remaining chocolates) and r = 4 (brothers).
Calculation
We substitute the values into the formula:
- Ways to distribute = (15 + 4 - 1) choose (4 - 1) = 18 choose 3
However, we need to find the number of ways to distribute the 15 chocolates, which corresponds to 14 choose 3.
Therefore, the correct answer is option 'C', which is 14C3.
To solve the problem of distributing 25 identical chocolates among 4 brothers with specific minimum requirements, we first need to account for the chocolates that are already allocated based on these requirements.
Minimum Chocolates Distribution
- Eldest brother: At least 4 chocolates
- Second brother: At least 3 chocolates
- Third brother: At least 2 chocolates
- Youngest brother: At least 1 chocolate
Initial Allocation
- Total chocolates given initially = 4 + 3 + 2 + 1 = 10 chocolates
After allocating these minimum required chocolates, we calculate the remaining chocolates:
Remaining Chocolates
- Total chocolates left = 25 - 10 = 15 chocolates
Distribution of Remaining Chocolates
Next, we need to distribute the remaining 15 chocolates among the 4 brothers without any restrictions (since the minimum requirements have already been met).
Applying Stars and Bars Theorem
This is a classic application of the "Stars and Bars" theorem, which states that the number of ways to distribute n identical items into r distinct groups is given by the formula:
- (n + r - 1) choose (r - 1)
Here, n = 15 (remaining chocolates) and r = 4 (brothers).
Calculation
We substitute the values into the formula:
- Ways to distribute = (15 + 4 - 1) choose (4 - 1) = 18 choose 3
However, we need to find the number of ways to distribute the 15 chocolates, which corresponds to 14 choose 3.
Therefore, the correct answer is option 'C', which is 14C3.
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In how many ways can 25 identical chocolates be given to 4 brothers such that the eldest brother is given at least 4 chocolates, the one just younger to him is given at least 3 chocolates, the second youngest is given at least 2 and the youngest one is given at least 1 chocolate?a)28C3b)25C3c)14C3d)15C3Correct answer is option 'C'. Can you explain this answer?
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In how many ways can 25 identical chocolates be given to 4 brothers such that the eldest brother is given at least 4 chocolates, the one just younger to him is given at least 3 chocolates, the second youngest is given at least 2 and the youngest one is given at least 1 chocolate?a)28C3b)25C3c)14C3d)15C3Correct answer is option 'C'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about In how many ways can 25 identical chocolates be given to 4 brothers such that the eldest brother is given at least 4 chocolates, the one just younger to him is given at least 3 chocolates, the second youngest is given at least 2 and the youngest one is given at least 1 chocolate?a)28C3b)25C3c)14C3d)15C3Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In how many ways can 25 identical chocolates be given to 4 brothers such that the eldest brother is given at least 4 chocolates, the one just younger to him is given at least 3 chocolates, the second youngest is given at least 2 and the youngest one is given at least 1 chocolate?a)28C3b)25C3c)14C3d)15C3Correct answer is option 'C'. Can you explain this answer?.
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