Answer :
To find the 4th term of the expansion of [tex]\((x-y)^{12}\)[/tex], we use the Binomial Theorem. The general term for the binomial expansion of [tex]\((x-y)^n\)[/tex] is given by:
[tex]\[
T_{k+1} = \binom{n}{k} x^{n-k} (-y)^k
\][/tex]
where:
- [tex]\( n = 12 \)[/tex] (the exponent in the binomial),
- [tex]\( k \)[/tex] is the term number minus 1 (so for the 4th term, [tex]\( k = 3 \)[/tex]),
- [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient.
Let's go through the steps:
1. Identify [tex]\( n \)[/tex] and [tex]\( k \)[/tex]:
- [tex]\( n = 12 \)[/tex]
- For the 4th term, [tex]\( k = 3 \)[/tex].
2. Calculate the binomial coefficient [tex]\( \binom{12}{3} \)[/tex]:
[tex]\[
\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
\][/tex]
3. Determine the powers of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- The power of [tex]\( x \)[/tex] is [tex]\( n-k = 12-3 = 9 \)[/tex].
- The power of [tex]\( y \)[/tex] is [tex]\( k = 3 \)[/tex].
4. Apply the sign for [tex]\((-y)^k\)[/tex]:
- Since [tex]\((-y)^3\)[/tex] results in a negative sign [tex]\((-1)^3 = -1\)[/tex], the term will be negative.
5. Combine everything to form the term:
[tex]\[
T_4 = 220 \cdot x^9 \cdot (-y)^3 = -220x^9y^3
\][/tex]
Therefore, the 4th term of the expansion of [tex]\((x-y)^{12}\)[/tex] is [tex]\(-220 x^9 y^3\)[/tex]. So, the correct answer is:
[tex]\[
-220 x^9 y^3
\][/tex]
[tex]\[
T_{k+1} = \binom{n}{k} x^{n-k} (-y)^k
\][/tex]
where:
- [tex]\( n = 12 \)[/tex] (the exponent in the binomial),
- [tex]\( k \)[/tex] is the term number minus 1 (so for the 4th term, [tex]\( k = 3 \)[/tex]),
- [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient.
Let's go through the steps:
1. Identify [tex]\( n \)[/tex] and [tex]\( k \)[/tex]:
- [tex]\( n = 12 \)[/tex]
- For the 4th term, [tex]\( k = 3 \)[/tex].
2. Calculate the binomial coefficient [tex]\( \binom{12}{3} \)[/tex]:
[tex]\[
\binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
\][/tex]
3. Determine the powers of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- The power of [tex]\( x \)[/tex] is [tex]\( n-k = 12-3 = 9 \)[/tex].
- The power of [tex]\( y \)[/tex] is [tex]\( k = 3 \)[/tex].
4. Apply the sign for [tex]\((-y)^k\)[/tex]:
- Since [tex]\((-y)^3\)[/tex] results in a negative sign [tex]\((-1)^3 = -1\)[/tex], the term will be negative.
5. Combine everything to form the term:
[tex]\[
T_4 = 220 \cdot x^9 \cdot (-y)^3 = -220x^9y^3
\][/tex]
Therefore, the 4th term of the expansion of [tex]\((x-y)^{12}\)[/tex] is [tex]\(-220 x^9 y^3\)[/tex]. So, the correct answer is:
[tex]\[
-220 x^9 y^3
\][/tex]
