Need help To Binomial Theorem And Binomial Coefficients

[Nipuna]

What are Binomial Theorem And Binomial Coefficients?

No Much Exploitations in VLE

[Herath]

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http://en.wikipedia.org/wiki/Binomial_theorem

[Neo]

Binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are non-negative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

I'll give you a quick guide.

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Say you have to expand the binomial (x + y)³.

Step 1: Write x terms from power 3 to 0.
x³ + x² + x¹ + x⁰

Step 2: Add y terms from power 0 to 3 after the x term (In reverse order).
x³y⁰ + x²y¹ + x¹y² + x⁰y³

Step 3: Remove power 0 terms (x⁰ = y⁰ = 1). Also write power 1 terms without power sysmbol (x1 = x, y1 = y)
x³ + x²y + xy² + y³

Step 4: Add coefficients (??????) according to Pascal Triangle for Power 3 (4th (3+1) raw) since we are expanding (x + y)³.
1 3 3 1 are the values from Pascal Triangle for Power 3 (4th raw). These are called binomial coefficients.
1x³ + 3x²y + 3xy² + 1y³

Finally we get:
(x + y)³ = x³ + 3x²y + 3xy² + y³

Let's evaluate (a + b)⁵ quickly now.
Step 1and 2 together:
a⁵b⁰ + a⁴b¹ + a³b² + a²b³ + a¹b⁴ + a⁰b⁵

Step 3:
a⁵ + a⁴b + a³b² + a²b³ + ab⁴ + b⁵

Step 4:
Chose the binomial coefficients from Pascal triangle. We need the value for Power 5. So we look at the 6th raw.
1 5 10 10 5 1
1a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + 1b⁵

Finally we have,
(a + b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

I guess you are now clear on the process.