To determine the number of possible groups of students that can fill the slots in Mr. Bentley's and Ms. Curren's classes, we need to use combinations.
1. Determine the number of combinations for Mr. Bentley's class:
- Mr. Bentley's class has 21 students, and we need to choose 5 students.
- The number of ways to choose 5 students out of 21 is given by the combination formula [tex]\( \binom{21}{5} \)[/tex].
- Using the combination formula, we find that [tex]\( \binom{21}{5} = 20,349 \)[/tex].
2. Determine the number of combinations for Ms. Curren's class:
- Ms. Curren's class has 23 students, and we need to choose 6 students.
- The number of ways to choose 6 students out of 23 is given by the combination formula [tex]\( \binom{23}{6} \)[/tex].
- Using the combination formula, we find that [tex]\( \binom{23}{6} = 100,947 \)[/tex].
3. Calculate the total number of possible groups:
- The total number of possible groups of students is the product of the number of combinations for Mr. Bentley's class and the number of combinations for Ms. Curren's class.
- Therefore, the total number of possible groups is [tex]\( 20,349 \)[/tex] multiplied by [tex]\( 100,947 \)[/tex], which results in [tex]\( 20,349 \cdot 100,947 \)[/tex].
Thus, the expression representing the number of possible groups of students that can fill the 11 available slots is:
[tex]\[ 20,349 \cdot 100,947 \][/tex]
