College

Find the 7th term of the expansion of [tex](3c + 2d)^9[/tex].

A. [tex]760c^3d^6[/tex]
B. [tex]760c^4d^5[/tex]
C. [tex]145,152c^3d^6[/tex]
D. [tex]145,152c^4d^5[/tex]

Please select the best answer from the choices provided.

A
B
C
D

Answer :

To find the 7th term in the expansion of [tex]\((3c + 2d)^9\)[/tex], we use the binomial theorem. The theorem tells us that the expansion of [tex]\((a + b)^n\)[/tex] can be expressed as:

[tex]\[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k. \][/tex]

Here, [tex]\(a = 3c\)[/tex], [tex]\(b = 2d\)[/tex], and [tex]\(n = 9\)[/tex].

The general term in the expansion is given by:

[tex]\[ T_{k+1} = \binom{n}{k} (3c)^{n-k} (2d)^k. \][/tex]

For the 7th term, we need [tex]\(k = 6\)[/tex] (since terms are numbered from [tex]\(k = 0\)[/tex]).

1. Calculate the Binomial Coefficient:
[tex]\[ \binom{9}{6} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84. \][/tex]

2. Calculate the powers:
- [tex]\((3c)^{9-6} = (3c)^3 = 27c^3\)[/tex],
- [tex]\((2d)^6 = 64d^6\)[/tex].

3. Combine these parts to find the term:
[tex]\[ T_7 = 84 \times 27c^3 \times 64d^6. \][/tex]

Now multiply these numbers:

- [tex]\(84 \times 27 = 2268\)[/tex],
- [tex]\(2268 \times 64 = 145152\)[/tex].

So, the 7th term is:

[tex]\[ 145152c^3d^6. \][/tex]

Therefore, the answer is c. [tex]\(145,152 c^3 d^6\)[/tex].