The fourth term in the Binomial expansion of [tex](e + 2f)^{10}[/tex] is [tex]10C_{3} (e)^{3}(2f)^{7}[/tex]
In elementary algebra, The Binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial [tex](x + y)^{n}[/tex] into a sum involving terms of the form [tex]ax^{b}y^{c}[/tex], where the exponents b and c are nonnegative integers with b + c = n , and the coefficient a of each term is a specific positive integer depending on n and b.
The binomial theorem formula is [tex](x + y)^{n}[/tex] = ∑ [tex]nC_{r} x^{n-r}y^{r}[/tex], where n is a positive integer and x, y are real numbers, and 0 < r ≤ n.
The formula to find the nth term in the binomial expansion of [tex](x + y)^{n}[/tex] is [tex]T_{r+1} = nC_{r} x^{n-r}y^{r}.[/tex]
As question demands fourth term of the expansion we need to substitute
r = 3 in the formula of nth term
On substituting we get
[tex]T_{3+1} = 10C_{3} (e)^{10-3}(2f)^{3}.[/tex]
[tex]T_{4} =[/tex] [tex]10C_{3} (e)^{3}(2f)^{7}[/tex]
Hence the fourth term in the binomial expansion of [tex](e + 2f)^{10}[/tex] is [tex]10C_{3} (e)^{3}(2f)^{7}[/tex]
Learn more about binomial theorem here :
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