Question

The 11th row of Pascal's triangle is shown below.

b- Hence find the coefficient of x³ in the expansion of (1 + 2x)10.

Answer 1

The coefficient of x³ in the expansion of (1 + 2x)¹⁰ is found using the binomial theorem. By applying the theorem, we identify the fourth term which corresponds to the coefficient of x³, resulting in the coefficient 1008.

Step-by-step explanation:

To find the coefficient of x³ in the expansion of (1 + 2x)¹⁰, we can use the binomial theorem. This theorem states that:

(a + b)⁹ = a⁹ + 9a⁸b + 36a⁷b² + 84a⁶b³ + ... + b⁹

The general term in the expansion of (a + b)⁹ is given by:

Tᵣ = [⁹Choose(n-r+1)] x a⁹⁻ᵣ x bᵣ⁻¹

Where n is 9 in this case, and r represents the term number.

We want to find the coefficient of x³, which correlates to the term with b³. This is the fourth term where r = 4:

T₄ = [⁹Choose4] x a⁶ x b³

Substituting a = 1 and b = 2x, we get:

T₄ = [⁹Choose4] x 1¶ x (2x)³

Calculating [⁹Choose4] which is 126 and simplifying, we find:

T₄ = 126 x 1¶ x 8x³ = 1008x³

Therefore, the coefficient of x³ in the expansion of (1 + 2x)¹⁰ is 1008.