The kindergarten teacher has 25 kid in her class. She takes 5 of them at a time, to zoological garden as often as she can, without taking the same 5 kids more than once. Then the number of visits the teacher makes to the garden exceeds that of a kid by

Options

A. `""^(25)C_(5)-""^(24)C_(5)`
B. `""^(24)C_(4)`
C. `""^(24)C_(5)`
D. `""^(25)C_(5)-""^(24)C_(4)`

Answer

To solve the problem step by step, we need to determine the number of visits the teacher makes to the zoological garden and the number of visits each kid makes. Then, we will find the difference between these two numbers.

### Step 1: Calculate the number of ways the teacher can select 5 kids from 25.
The number of ways to choose 5 kids from 25 can be calculated using the combination formula:

\[ \text{Number of visits by teacher} = \binom{25}{5} \]

### Step 2: Calculate the number of visits each kid makes.
Each kid can be part of a group of 5 kids. If one kid has already visited the zoo, we need to choose the remaining 4 kids from the other 24 kids. Thus, the number of visits each kid can make is:

\[ \text{Number of visits by each kid} = \binom{24}{4} \]

### Step 3: Find the difference between the number of visits by the teacher and the number of visits by each kid.
Now, we need to find the difference between the total visits made by the teacher and the visits made by each kid:

\[ \text{Difference} = \binom{25}{5} - \binom{24}{4} \]

### Step 4: Calculate the values of the combinations.
Now, we will calculate the values of the combinations:

1. Calculate \(\binom{25}{5}\):
\[ \binom{25}{5} = \frac{25!}{5!(25-5)!} = \frac{25!}{5! \cdot 20!} = \frac{25 \times 24 \times 23 \times 22 \times 21}{5 \times 4 \times 3 \times 2 \times 1} = 53130 \]

2. Calculate \(\binom{24}{4}\):
\[ \binom{24}{4} = \frac{24!}{4!(24-4)!} = \frac{24!}{4! \cdot 20!} = \frac{24 \times 23 \times 22 \times 21}{4 \times 3 \times 2 \times 1} = 10626 \]

### Step 5: Find the final result.
Now we can substitute the calculated values back into the difference:

\[ \text{Difference} = 53130 - 10626 = 42504 \]

Thus, the number of visits the teacher makes to the garden exceeds that of a kid by **42504**.