From a class of 40 students, in how many ways can five students be chosen
(i) For an excursion party.

Answer

To solve the problem of choosing 5 students from a class of 40 students for an excursion party, we can use the concept of combinations. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We need to choose 5 students from a total of 40 students. The order in which we choose the students does not matter, so we will use combinations.

### Step 2: Use the Combination Formula
The number of ways to choose \( r \) items from \( n \) items is given by the combination formula:
\[ nCr = \frac{n!}{r!(n-r)!} \]
In our case, \( n = 40 \) and \( r = 5 \).

### Step 3: Substitute Values into the Formula
Substituting the values into the formula, we get:
\[ 40C5 = \frac{40!}{5!(40-5)!} = \frac{40!}{5! \cdot 35!} \]

### Step 4: Simplify the Factorial Expression
We can simplify \( 40! \) as follows:
\[ 40! = 40 \times 39 \times 38 \times 37 \times 36 \times 35! \]
Thus, we can rewrite the combination as:
\[ 40C5 = \frac{40 \times 39 \times 38 \times 37 \times 36 \times 35!}{5! \cdot 35!} \]
The \( 35! \) in the numerator and denominator cancels out:
\[ 40C5 = \frac{40 \times 39 \times 38 \times 37 \times 36}{5!} \]

### Step 5: Calculate \( 5! \)
Now, calculate \( 5! \):
\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

### Step 6: Substitute and Calculate
Now, we substitute \( 5! \) back into the equation:
\[ 40C5 = \frac{40 \times 39 \times 38 \times 37 \times 36}{120} \]

### Step 7: Calculate the Numerator
Calculating the numerator:
\[ 40 \times 39 = 1560 \]
\[ 1560 \times 38 = 59280 \]
\[ 59280 \times 37 = 2193360 \]
\[ 2193360 \times 36 = 78981760 \]

### Step 8: Divide by \( 120 \)
Now, divide by \( 120 \):
\[ 40C5 = \frac{78981760}{120} = 658180 \]

### Final Answer
Thus, the number of ways to choose 5 students from a class of 40 is:
\[ \boxed{658008} \]