Binomial Theorem
\((a + b)^{2}\) ya \((a+b)^{3}\) to hum haath se expand kar lete hain, par \((a+b)^{12}\) ko baar-baar multiply karna bahut lamba kaam hai. Binomial theorem ek formula deta hai jisse \((a+b)^{n}\) ko seedha expand kar sakte hain — bina multiply kiye.
1. Pattern: Pascal's Triangle
\((a+b)^{n}\) ke coefficients ek triangle banate hain, jahan har number uske upar wale do numbers ka sum hota hai:
\[ \begin{array}{c} 1 \\ 1 \quad 1 \\ 1 \quad 2 \quad 1 \\ 1 \quad 3 \quad 3 \quad 1 \\ 1 \quad 4 \quad 6 \quad 4 \quad 1 \end{array} \]
In coefficients ko hum \({}^{n}C_{r}\) (combinations) se bhi likh sakte hain.
2. Binomial Theorem (Positive Integer \(n\))
Kisi positive integer \(n\) ke liye: \[ (a + b)^{n} = \sum_{r=0}^{n} {}^{n}C_{r}\, a^{\,n-r}\, b^{\,r}. \]
Khol kar:
\[ (a+b)^{n} = {}^{n}C_{0}a^{n} + {}^{n}C_{1}a^{n-1}b + {}^{n}C_{2}a^{n-2}b^{2} + \dots + {}^{n}C_{n}b^{n}. \]
Expansion ki khaas baatein
- Total \(n + 1\) terms hote hain.
- Har term me \(a\) aur \(b\) ke powers ka sum hamesha \(n\) hota hai.
- \(a\) ka power ghatta hai (\(n\) se \(0\)), \(b\) ka badhta hai (\(0\) se \(n\)).
- Coefficients dono sire se symmetric hote hain (\({}^{n}C_{r} = {}^{n}C_{n-r}\)).
\((x + 2)^{4}\) expand karo. \[ = {}^{4}C_{0}x^{4} + {}^{4}C_{1}x^{3}(2) + {}^{4}C_{2}x^{2}(2)^{2} + {}^{4}C_{3}x(2)^{3} + {}^{4}C_{4}(2)^{4} \] \[ = x^{4} + 8x^{3} + 24x^{2} + 32x + 16. \]
3. General Term
\((a+b)^{n}\) ka general term (\((r+1)\)-th term): \[ T_{r+1} = {}^{n}C_{r}\, a^{\,n-r}\, b^{\,r}. \]
Isse koi bhi particular term ya "coefficient of \(x^{k}\)" type ke sawaal aasaani se nikalte hain.
\((2x - 3)^{6}\) ke expansion me 4th term nikaalo. Yahan \(r + 1 = 4 \Rightarrow r = 3\), \(a = 2x\), \(b = -3\), \(n = 6\): \[ T_{4} = {}^{6}C_{3}(2x)^{3}(-3)^{3} = 20 \cdot 8x^{3} \cdot (-27) = -4320\,x^{3}. \]
4. Middle Term
- Agar \(n\) even ho, to ek hi middle term: \(\left(\dfrac{n}{2} + 1\right)\)-th term.
- Agar \(n\) odd ho, to do middle terms: \(\dfrac{n+1}{2}\)-th aur \(\dfrac{n+3}{2}\)-th.
5. Useful Special Cases
\(a = b = 1\) daalne par:
\[ {}^{n}C_{0} + {}^{n}C_{1} + {}^{n}C_{2} + \dots + {}^{n}C_{n} = 2^{n}. \]
Yaani kisi set ke saare subsets ki ginti \(2^{n}\) hoti hai (Sets chapter se connection!).
Key Takeaways
- \((a+b)^{n} = \displaystyle\sum_{r=0}^{n} {}^{n}C_{r}\,a^{n-r}b^{r}\); total \(n+1\) terms.
- General term \(T_{r+1} = {}^{n}C_{r}\,a^{n-r}b^{r}\) — har particular-term sawaal ki chaabi.
- Powers ka sum hamesha \(n\); coefficients symmetric.
- Middle term \(n\) even/odd par depend karta hai.
- \(\sum {}^{n}C_{r} = 2^{n}\).