To determine the third term of the binomial expansion [tex]\((x + y)^4\)[/tex], we can use the binomial theorem. The binomial theorem states that:
[tex]\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \][/tex]
Here, [tex]\(a = x\)[/tex], [tex]\(b = y\)[/tex], and [tex]\(n = 4\)[/tex]. We need to find the third term of this expansion. The general term for the binomial expansion is given by:
[tex]\[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \][/tex]
To find the third term ([tex]\(k = 2\)[/tex]), we substitute [tex]\(n = 4\)[/tex], [tex]\(a = x\)[/tex], [tex]\(b = y\)[/tex], and [tex]\(k = 2\)[/tex] into the formula:
1. Calculate the binomial coefficient [tex]\(\binom{4}{2}\)[/tex]:
[tex]\[ \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1} = \frac{24}{4} = 6 \][/tex]
2. Substitute [tex]\(k = 2\)[/tex] into the term formula:
[tex]\[ T_{3} = \binom{4}{2} x^{4-2} y^2 = 6 x^2 y^2 \][/tex]
Thus, the third term in the binomial expansion of [tex]\((x + y)^4\)[/tex] is:
[tex]\[ 6 x^2 y^2 \][/tex]
### Solution
The binomial expansion of [tex]\((x + y)^4\)[/tex] is:
[tex]\[ (x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4 \][/tex]
And as per the detailed steps above, the third term is:
[tex]\[ 6 x^2 y^2 \][/tex]
