Binomial Theorem | द्विपद प्रमेय

Overview | अध्याय परिचय

Binomial Theorem deals with the expansion of expressions raised to a power. It helps simplify expressions like (a + b)ⁿ.
द्विपद प्रमेय (Binomial Theorem) किसी द्विपदीय व्यंजक (expression) को किसी घातांक पर उठाने पर उसका विस्तार बताता है।

Key Formula | मुख्य सूत्र

General expansion of (a + b)ⁿ:
(a + b)ⁿ = C(n, 0)·aⁿ + C(n, 1)·aⁿ⁻¹·b + C(n, 2)·aⁿ⁻²·b² + ... + C(n, n)·bⁿ

जहाँ C(n, r) = n! / [r!(n − r)!] को संयोजन (Combination) कहते हैं।

Tips & Tricks | युक्तियाँ

• Number of terms = n + 1
• Middle term(s): For even n, single middle term; for odd n, two middle terms.
• Use symmetrical property: C(n, r) = C(n, n − r)
• Term independent of variable means variable's power is 0.

• पदों की संख्या = n + 1
• मध्य पद: n सम हो तो एक, विषम हो तो दो मध्य पद होंगे।
• सममित गुण: C(n, r) = C(n, n − r)
• यदि किसी पद में चर (variable) अनुपस्थित है, तो उस पद की घातांक 0 होती है।

Hint-based Questions | संकेत-आधारित प्रश्न

1. Find the 5th term in the expansion of (2x + 3)⁶.
   संकेत: Use T_r+1 = C(n, r)·a^n−r·b^r
2. Find the term independent of x in the expansion of (x² + 1/x)⁶.
   संकेत: Find r such that the power of x becomes 0.
3. Find the middle term(s) in the expansion of (1 − x)¹⁰.
   संकेत: n = 10 → single middle term is (n/2 + 1)th = 6th term
4. Find the coefficient of x³ in (2 + x)⁶.
   संकेत: Use the expansion and identify the term with x³
5. Find the 3rd term from the end in the expansion of (x − 3)⁵.
   संकेत: Use symmetry: (rth from end) = (n − r + 2)th term

Practice Questions | अभ्यास प्रश्न

1. Expand (x + 1)⁵ and find the coefficient of x³.
2. Find the general term in the expansion of (a − b)⁸.
3. Find the middle term(s) in the expansion of (1 + x)⁹.
4. In (2 − x)⁷, find the term containing x⁴.
5. Find the coefficient of x⁻² in (x + 1/x)⁶.
6. Which term is independent of x in the expansion of (2x − 1/x)⁸?