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If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then r is
- a)7
- b)8
- c)9
- d)10
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)t...
Understanding the Binomial Expansion
In the expansion of (1 + x)^20, each term can be represented using the binomial theorem. The r-th term in the expansion is given by the formula:
- T(r) = C(20, r-1) * x^(r-1)
Where C(n, k) denotes the binomial coefficient “n choose k”.
Condition for Equal Coefficients
The problem states that the coefficients of the r-th term and the (r + 4)-th term are equal. Thus, we can set up the equation:
- C(20, r-1) = C(20, (r + 4) - 1)
This simplifies to:
- C(20, r-1) = C(20, r + 3)
Using the Property of Binomial Coefficients
The property of binomial coefficients states that C(n, k) = C(n, n-k). Applying this, we have:
- C(20, r-1) = C(20, 20 - (r + 3))
This leads to:
- C(20, r-1) = C(20, 17 - r)
Setting the indices equal gives us the equation:
- r - 1 = 17 - r
Solving for r
Rearranging this equation leads to:
- 2r = 18
- r = 9
Conclusion
Thus, the value of r for which the coefficients of the r-th and (r + 4)-th terms are equal is:
- r = 9
Hence, the correct answer is option 'C'.
In the expansion of (1 + x)^20, each term can be represented using the binomial theorem. The r-th term in the expansion is given by the formula:
- T(r) = C(20, r-1) * x^(r-1)
Where C(n, k) denotes the binomial coefficient “n choose k”.
Condition for Equal Coefficients
The problem states that the coefficients of the r-th term and the (r + 4)-th term are equal. Thus, we can set up the equation:
- C(20, r-1) = C(20, (r + 4) - 1)
This simplifies to:
- C(20, r-1) = C(20, r + 3)
Using the Property of Binomial Coefficients
The property of binomial coefficients states that C(n, k) = C(n, n-k). Applying this, we have:
- C(20, r-1) = C(20, 20 - (r + 3))
This leads to:
- C(20, r-1) = C(20, 17 - r)
Setting the indices equal gives us the equation:
- r - 1 = 17 - r
Solving for r
Rearranging this equation leads to:
- 2r = 18
- r = 9
Conclusion
Thus, the value of r for which the coefficients of the r-th and (r + 4)-th terms are equal is:
- r = 9
Hence, the correct answer is option 'C'.
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If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then r isa)7b)8c)9d)10Correct answer is option 'C'. Can you explain this answer?
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If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then r isa)7b)8c)9d)10Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then r isa)7b)8c)9d)10Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then r isa)7b)8c)9d)10Correct answer is option 'C'. Can you explain this answer?.
If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then r isa)7b)8c)9d)10Correct answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then r isa)7b)8c)9d)10Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If in the expansion of (1 + x)20, the coefficients of rth and (r + 4)th terms are equal, then r isa)7b)8c)9d)10Correct answer is option 'C'. Can you explain this answer?.
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