\begin{tabular}{l} Answer the questions below. \ $$ \begin{array}{l} ext { (a) There are } 15 ext { European cities that Juan would eventually like to visit. On his next vacation, } \ ext { though, he only has time to visit } 4 ext { of the cities: one on Monday, one on Tuesday, one on Wednesday, } \ ext { and one on Thursday. He is now trying to make a schedule of which city he'll visit on which day. How } \ ext { many different schedules are possible? (Assume that he will not visit a city more than once.) } \ \square\end{array} \begin{array}{l} ext { (b) To log on to a certain computer account, the user must type in a 4-letter password. In such a } \ ext { password, no letter may be repeated, and only the lower case of a letter may be used. How many } \ ext { such 4-letter passwords are possible? (There are } 26 ext { letters in the alphabet.) } \ \square\end{array} $$ \ \hline\end{tabular}

Question

\begin{tabular}{l} Answer the questions below. \ $$ \begin{array}{l}	ext { (a) There are } 15 	ext { European cities that Juan would eventually like to visit. On his next vacation, } \ 	ext { though, he only has time to visit } 4 	ext { of the cities: one on Monday, one on Tuesday, one on Wednesday, } \ 	ext { and one on Thursday. He is now trying to make a schedule of which city he'll visit on which day. How } \ 	ext { many different schedules are possible? (Assume that he will not visit a city more than once.) } \ \square\end{array} \begin{array}{l}	ext { (b) To log on to a certain computer account, the user must type in a 4-letter password. In such a } \ 	ext { password, no letter may be repeated, and only the lower case of a letter may be used. How many } \ 	ext { such 4-letter passwords are possible? (There are } 26 	ext { letters in the alphabet.) } \ \square\end{array} $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

Let's solve each part step by step.

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### **Problem (a)**

**Question:**
Juan has 15 European cities he wants to visit and plans to visit 4 of them over four consecutive days (Monday to Thursday). He will visit a different city each day. How many different schedules are possible?

**Solution:**
We need to determine the number of ways to assign 4 distinct cities to 4 distinct days.

This is a permutation problem since the order in which the cities are visited matters (each day is distinct).

The number of permutations of 15 cities taken 4 at a time is calculated as:

$$
P(15, 4) = 15 \times 14 \times 13 \times 12
$$

**Calculation:**

$$
P(15, 4) = 15 \times 14 \times 13 \times 12 = 32,\!760
$$

**Answer:**
There are **32,760** different possible schedules.

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### **Problem (b)**

**Question:**
To log on to a certain computer account, the user must enter a 4-letter password. Each letter must be unique, and only lowercase letters (a–z) are allowed. How many such 4-letter passwords are possible?

**Solution:**
We need to determine the number of 4-letter arrangements where no letter is repeated.

Since only lowercase letters are used and there are 26 letters in the alphabet, the number of possible passwords is the number of permutations of 26 letters taken 4 at a time.

$$
P(26, 4) = 26 \times 25 \times 24 \times 23
$$

**Calculation:**

$$
P(26, 4) = 26 \times 25 \times 24 \times 23 = 358,\!800
$$

**Answer:**
There are **358,800** possible 4-letter passwords.

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There are 32,760 different schedules possible for visiting the cities and 358,800 possible 4-letter passwords.

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