I see you uploaded a PDF with several mathematical problems. I'll go through each question one by one, providing a detailed solution.

1. Use the binomial theorem to expand $(a + 2)^3 \cdot (3a - 2b)^0$

Since $(3a - 2b)^0 = 1$, we only need to expand $(a + 2)^3$.

Using the binomial theorem: $(a + 2)^3 = \sum_{k=0}^{3} \binom{3}{k} a^{3-k} \cdot 2^k$ We expand: $= \binom{3}{0}a^3(2^0) + \binom{3}{1}a^2(2^1) + \binom{3}{2}a(2^2) + \binom{3}{3}(2^3)$

$= 1 \cdot a^3 \cdot 1 + 3 \cdot a^2 \cdot 2 + 3 \cdot a \cdot 4 + 1 \cdot 8$ $= a^3 + 6a^2 + 12a + 8$

Thus, the expansion is: $(a + 2)^3 = a^3 + 6a^2 + 12a + 8$

2. Find the eighth term in the expansion of $(2a - 3b)^9$

The general term in the binomial expansion is given by: $T_{k+1} = \binom{n}{k} (2a)^{n-k} (-3b)^k$ Here, $n = 9$ and we need the 8th term, so $k = 7$.

Substitute $n = 9$ and $k = 7$: $T_8 = \binom{9}{7} (2a)^{9-7} (-3b)^7$ $= \binom{9}{7} (2a)^2 (-3b)^7$ $= \binom{9}{7} \cdot 4a^2 \cdot (-2187b^7)$ $= 36 \cdot 4a^2 \cdot (-2187b^7)$ $= -314928 a^2 b^7$

So, the eighth term is: $T_8 = -314928 a^2 b^7$

3. License plates containing 3 English letters followed by 4 digits

• How many license plates are possible?

Each letter can be chosen from 26 English letters, and each digit from 10 possible digits (0–9). Repetition is allowed.

So, the total number of possible plates is: $26^3 \times 10^4 = 17576 \times 10000 = 175760000$

Thus, there are 175,760,000 possible license plates.

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Let me know if you need further clarification or additional details on these questions before I proceed to the next set. Would you like to continue solving the next questions?

Here are some related questions that you may find interesting:

1. How does the binomial theorem apply to different types of algebraic expressions?
2. What are common patterns in binomial expansions for higher powers?
3. What are some applications of binomial expansion in probability and statistics?
4. How do you calculate the number of combinations for letters and digits in a different plate format?
5. What are other applications of combination formulas outside of license plates?

Tip: Always check if a binomial expression involves zero or one power terms before expanding to simplify the process.