To find the number of ways to choose 5 students from a class of 17 students, we can use the combination formula, which is represented as:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of students (17 in this case), \( r \) is the number of students to choose (5 in this case), and \( ! \) denotes factorial.
Using this formula:
\[ \binom{17}{5} = \frac{17!}{5!(17-5)!} = \frac{17!}{5! \cdot 12!} \]
Calculating this step-by-step:
- Calculate \( 17! \) up to \( 12! \):
\[ \frac{17 \times 16 \times 15 \times 14 \times 13 \times 12!}{5! \times 12!} \]
- The \( 12! \) terms cancel out:
\[ \frac{17 \times 16 \times 15 \times 14 \times 13}{5!} \]
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Now calculate \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
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Now compute the product of the numerator:
\[ 17 \times 16 = 272, \] \[ 272 \times 15 = 4080, \] \[ 4080 \times 14 = 57120, \] \[ 57120 \times 13 = 742560. \]
- Finally, divide by \( 120 \):
\[ \frac{742560}{120} = 6188. \]
So the number of ways to choose 5 students from a class of 17 is:
\[ \binom{17}{5} = 6188. \]
Therefore, the correct answer is 6,188.