Binomial Theorem ની ગણતરી કેવી રીતે કરવી
Binomial Theorem શું છે?
The binomial theorem describes how to expand expressions of the form (a + b)ⁿ. It uses Pascal's triangle coefficients and is fundamental to algebra, probability, and combinatorics.
સૂત્ર
(a + b)ⁿ = Σₖ₌₀ⁿ C(n,k) aⁿ⁻ᵏ bᵏ where C(n,k) = n!/(k!(n−k)!)
- n
- exponent — non-negative integer
- a, b
- terms being expanded
- C(n,k)
- binomial coefficient — "n choose k"
પગલું દ્વારા પગલું માર્ગદર્શિકા
- 1(a + b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ
- 2Coefficients C(n,k) = n! / (k!(n−k)!)
- 3Coefficients form Pascal's triangle
- 4(a+b)² = a² + 2ab + b²
Worked Examples
ઇનપુટ
(a+b)³
પરિણામ
a³ + 3a²b + 3ab² + b³
ઇનપુટ
(x+1)⁴
પરિણામ
x⁴ + 4x³ + 6x² + 4x + 1
Frequently Asked Questions
What is Pascal's triangle and how does it relate to binomial expansion?
Pascal's triangle lists the binomial coefficients. Each row n gives the coefficients for (a+b)ⁿ.
Can binomial expansion be used for negative exponents?
Yes, the binomial series extends to any real exponent, but for negative/fractional exponents it's an infinite series.
What is the multinomial theorem?
A generalization of binomial expansion to three or more terms: (a+b+c)ⁿ.
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