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The coefficients of xp and xq (p, q are + ve integers) in the binomial expansion of (1+x)p+q are
- a)equal
- b)reciprocal of each other
- c)equal numerically
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Understanding the Binomial Expansion
The binomial expansion of (1 + x)^(p + q) can be analyzed to find the coefficients of specific terms, namely xp and xq.
Coefficients in the Expansion
- The general term in the expansion of (1 + x)^(n) is given by:
T(k) = C(n, k) * x^k,
where C(n, k) is the binomial coefficient representing "n choose k".
- For our case, n = p + q, and we want to find the coefficients of xp and xq.
Finding the Coefficient of xp
- The coefficient of xp in (1 + x)^(p + q) occurs when k = p.
Therefore, the coefficient is C(p + q, p).
Finding the Coefficient of xq
- Similarly, the coefficient of xq occurs when k = q.
Thus, the coefficient is C(p + q, q).
Relationship Between Coefficients
- By the properties of binomial coefficients, we know:
C(n, k) = C(n, n - k).
This implies:
C(p + q, p) = C(p + q, q).
- Hence, the coefficients of xp and xq are equal numerically.
Conclusion
- The coefficients of xp and xq in the binomial expansion of (1 + x)^(p + q) are equal.
- Therefore, the correct answer is option 'A': they are equal.
The binomial expansion of (1 + x)^(p + q) can be analyzed to find the coefficients of specific terms, namely xp and xq.
Coefficients in the Expansion
- The general term in the expansion of (1 + x)^(n) is given by:
T(k) = C(n, k) * x^k,
where C(n, k) is the binomial coefficient representing "n choose k".
- For our case, n = p + q, and we want to find the coefficients of xp and xq.
Finding the Coefficient of xp
- The coefficient of xp in (1 + x)^(p + q) occurs when k = p.
Therefore, the coefficient is C(p + q, p).
Finding the Coefficient of xq
- Similarly, the coefficient of xq occurs when k = q.
Thus, the coefficient is C(p + q, q).
Relationship Between Coefficients
- By the properties of binomial coefficients, we know:
C(n, k) = C(n, n - k).
This implies:
C(p + q, p) = C(p + q, q).
- Hence, the coefficients of xp and xq are equal numerically.
Conclusion
- The coefficients of xp and xq in the binomial expansion of (1 + x)^(p + q) are equal.
- Therefore, the correct answer is option 'A': they are equal.
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The coefficients ofxp andxq (p, q are + ve integers) in the binomial expansion of(1+x)p+q area)equalb)reciprocal of each otherc)equal numericallyd)None of theseCorrect answer is option 'A'. Can you explain this answer?
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