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Find the value of r, if the coefficients of (2r + 4)th and (r – 2)th terms in the expansion of (1 + x)18 are equal.
- a)6
- b)5
- c)4
- d)3
Correct answer is option 'A'. Can you explain this answer?
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Find the value of r, if the coefficients of (2r + 4)th and (r – ...
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Find the value of r, if the coefficients of (2r + 4)th and (r – ...
Understanding the Problem
We need to find the value of r such that the coefficients of the (2r + 4)th term and the (r - 2)th term in the expansion of (1 + x)^18 are equal.
Expansion of Binomial Expression
The general term (T_k) in the expansion of (1 + x)^n is given by:
- T_k = C(n, k-1) * x^(k-1)
Where C(n, k-1) is the binomial coefficient.
Identifying the Terms
For our case, with n = 18:
- Coefficient of (2r + 4)th term: C(18, 2r + 4 - 1) = C(18, 2r + 3)
- Coefficient of (r - 2)th term: C(18, r - 2 - 1) = C(18, r - 3)
Setting the Coefficients Equal
We set these coefficients equal to each other:
- C(18, 2r + 3) = C(18, r - 3)
Using the property of binomial coefficients, we know that:
- C(n, k) = C(n, n-k)
This gives us:
- 2r + 3 = 18 - (r - 3)
Simplifying the Equation
Now, we simplify the equation:
1. 2r + 3 = 18 - r + 3
2. 2r + r = 18 + 3 - 3
3. 3r = 18
4. r = 6
Conclusion
Thus, the value of r that satisfies the condition is:
- r = 6
Therefore, the correct answer is option 'A'.
We need to find the value of r such that the coefficients of the (2r + 4)th term and the (r - 2)th term in the expansion of (1 + x)^18 are equal.
Expansion of Binomial Expression
The general term (T_k) in the expansion of (1 + x)^n is given by:
- T_k = C(n, k-1) * x^(k-1)
Where C(n, k-1) is the binomial coefficient.
Identifying the Terms
For our case, with n = 18:
- Coefficient of (2r + 4)th term: C(18, 2r + 4 - 1) = C(18, 2r + 3)
- Coefficient of (r - 2)th term: C(18, r - 2 - 1) = C(18, r - 3)
Setting the Coefficients Equal
We set these coefficients equal to each other:
- C(18, 2r + 3) = C(18, r - 3)
Using the property of binomial coefficients, we know that:
- C(n, k) = C(n, n-k)
This gives us:
- 2r + 3 = 18 - (r - 3)
Simplifying the Equation
Now, we simplify the equation:
1. 2r + 3 = 18 - r + 3
2. 2r + r = 18 + 3 - 3
3. 3r = 18
4. r = 6
Conclusion
Thus, the value of r that satisfies the condition is:
- r = 6
Therefore, the correct answer is option 'A'.
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Find the value of r, if the coefficients of (2r + 4)th and (r – 2)th terms in the expansion of (1 + x)18are equal.a)6b)5c)4d)3Correct answer is option 'A'. Can you explain this answer?
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