A father has 2 apples and 3 pears. Each weekday (Monday through Friday) he gives one of the fruits to his daughter. In how many ways can this be done?

Options

A. 120
B. 10
C. 24
D. 12

Answer

To solve the problem of how many ways a father can give his daughter 2 apples and 3 pears over 5 weekdays, we can use the concept of permutations of a multiset.

### Step-by-Step Solution:

1. **Identify the total number of fruits**: The father has a total of 2 apples and 3 pears. Therefore, the total number of fruits is:
\[ \text{Total fruits} = 2 \text{ (apples)} + 3 \text{ (pears)} = 5 \text{ fruits} \]

**Hint**: Count the total number of items you have.

2. **Determine the arrangement of fruits**: We need to find the different ways to arrange 2 apples (A) and 3 pears (P). The arrangement can be represented as a sequence of letters, for example, AAPPP.

3. **Use the formula for permutations of a multiset**: The number of distinct arrangements of n items where there are groups of indistinguishable items is given by:
\[ \frac{n!}{n_1! \times n_2!} \]
where \( n \) is the total number of items, \( n_1 \) is the number of indistinguishable items of type 1, and \( n_2 \) is the number of indistinguishable items of type 2.

In this case:
- \( n = 5 \) (total fruits)
- \( n_1 = 2 \) (apples)
- \( n_2 = 3 \) (pears)

So the formula becomes:
\[ \text{Number of ways} = \frac{5!}{2! \times 3!} \]

**Hint**: Remember to use factorials to count arrangements when dealing with indistinguishable items.

4. **Calculate the factorials**:
- \( 5! = 120 \)
- \( 2! = 2 \)
- \( 3! = 6 \)

Now substitute these values into the formula:
\[ \text{Number of ways} = \frac{120}{2 \times 6} = \frac{120}{12} = 10 \]

**Hint**: Factorials grow quickly, so calculate them step by step.

5. **Conclusion**: The total number of ways the father can give the fruits to his daughter over the 5 weekdays is 10.

**Final Answer**: The answer is 10.