3) How many codes of 8 letters either begin with an $$ u $$ or end with 2 t's? ( 5 points)

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To solve the problem, we need to find the total number of 8-letter codes that either begin with the letter $$ u $$ or end with two $$ t $$'s. We will use the principle of inclusion-exclusion to account for any overlap between the two conditions.

1. **Codes that begin with $$ u $$**:
   - If the first letter is $$ u $$, we have 7 remaining positions to fill.
   - Each of these positions can be filled with any letter from the alphabet. Assuming we have 26 letters in the alphabet, the number of codes that begin with $$ u $$ is:
   $$
   26^7
   $$

2. **Codes that end with $$ tt $$**:
   - If the last two letters are $$ tt $$, we have 6 remaining positions to fill.
   - Each of these positions can also be filled with any letter from the alphabet. Thus, the number of codes that end with $$ tt $$ is:
   $$
   26^6
   $$

3. **Codes that begin with $$ u $$ and end with $$ tt $$**:
   - If the first letter is $$ u $$ and the last two letters are $$ tt $$, we have 5 remaining positions to fill.
   - Each of these positions can be filled with any letter from the alphabet. Therefore, the number of codes that both begin with $$ u $$ and end with $$ tt $$ is:
   $$
   26^5
   $$

Now, we can apply the principle of inclusion-exclusion to find the total number of codes that either begin with $$ u $$ or end with $$ tt $$:
$$
\text{Total} = (\text{Codes that begin with } u) + (\text{Codes that end with } tt) - (\text{Codes that begin with } u \text{ and end with } tt)
$$
Substituting the values we calculated:
$$
\text{Total} = 26^7 + 26^6 - 26^5
$$

Now, we can factor out $$ 26^5 $$:
$$
\text{Total} = 26^5 (26^2 + 26 - 1)
$$
Calculating $$ 26^2 + 26 - 1 $$:
$$
26^2 = 676, \quad 26^2 + 26 = 676 + 26 = 702, \quad 702 - 1 = 701
$$
Thus, we have:
$$
\text{Total} = 26^5 \times 701
$$

Finally, the total number of 8-letter codes that either begin with $$ u $$ or end with $$ tt $$ is:
$$
\boxed{26^5 \times 701}
$$
The total number of 8-letter codes that either begin with $$ u $$ or end with $$ tt $$ is $$ 26^5 \times 701 $$.

How many codes of 8 letters either begin with an or end with 2 t’s? ( 5 points)

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