Find the 4^th term from the last in the expansion of .

Concept:

Binomial Expansion:

• (a + b)n = C0 a0 bn + C1 a1 bn-1 + C2 a2 bn-2 + … + Cr ar bn-r + … + Cn-1 an-1 b1 + Cn an b0, where C0, C1, …, Cn are the Binomial Coefficients defined as Cr = nCr = .
• The total number of terms in the expansion is n + 1.
• The (r + 1)^th term in the expansion is T_r+1 = C_r a^r b^n-r.

Calculation:

For the given expression , n = 7.

There will be 7 + 1 = 8 terms in the expansion of the above expression.

Expanding by reversing the order of the terms (i.e. a = -3y and b = ), we have:

T₅ = T₄₊₁ = ⁷C₄  (-3y)⁴  = 35  (3⁴y⁴)  = 105x³y⁴.