Q. 46

# Find the coeffici

To Find : Coefficients of x4

For (1+x)n

a=1, b=x

We have a formula,

For (1-x)n

a=1, b=-x and n=n

We have formula,

Coefficients of x4 are

x0.x4 = C0C4

x1.x3 = - C1C3

x2.x2 = C2C2

x3.x1= - C3C1

x4.x0 = C4C0

Therefore, Coefficient of x4

= C4C0 - C1C3 + C2C2 - C3C1 + C4C0

Let us assume, n=4, it becomes

4C4 4C0 - 4C1 4C3 + 4C2 4C2 - 4C3 4C1 + 4C4 4C0

We know that ,

By using above formula, we get,

4C4 4C0 - 4C1 4C3 + 4C2 4C2 - 4C3 4C1 + 4C4 4C0

= (1)(1) – (4)(4) + (6)(6) – (4)(4) + (1)(1)

= 1 – 16 + 36 – 16 + 1

= 6

= 4C2

Therefore, in general,

C4C0 - C1C3 + C2C2 - C3C1 + C4C0 = C2

Therefore, Coefficient of x4 = C2

Conclusion :

• Coefficient of x4 = C2

• C4C0 - C1C3 + C2C2 - C3C1 + C4C0 = C2

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