To determine how many different ways the instructor can choose 4 students from 10, we use the combination formula, which is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n - r)!} \]
where \( n \) is the total number of items to choose from (in this case, 10 students), and \( r \) is the number of items to choose (in this case, 4 students).
So, we can calculate \( \binom{10}{4} \) as follows:
\[ \binom{10}{4} = \frac{10!}{4!(10 - 4)!} = \frac{10!}{4!6!} \]
Next, we can calculate \( 10! \), \( 4! \), and \( 6! \):
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)
Now we only need the first 4 factors of \( 10! \) divided by \( 6! \):
\[ 10! = 10 \times 9 \times 8 \times 7 \times 6! \]
So:
\[ \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4! \times 6!} = \frac{10 \times 9 \times 8 \times 7}{4!} \] \[ = \frac{10 \times 9 \times 8 \times 7}{24} \]
Calculating the numerator:
\[ 10 \times 9 = 90 \] \[ 90 \times 8 = 720 \] \[ 720 \times 7 = 5040 \]
Now divide by \( 24 \):
\[ \frac{5040}{24} = 210 \]
Thus, the number of different ways the instructor can choose 4 students from 10 is 210.
So the answer is:
\[ \boxed{210} \]
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