a high school is hosting a blood drive. the sign - up sheet has 5 slots available for mr. bentleys class and…

Explanation:

Step1: Calculate combinations for Mr. Bentley's class

We use the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 21$ (number of students in Mr. Bentley's class) and $r = 5$ (number of slots for Mr. Bentley's class). So $C(21,5)=\frac{21!}{5!(21 - 5)!}=\frac{21!}{5!×16!}=\frac{21\times20\times19\times18\times17}{5\times4\times3\times2\times1}=20349$.

Step2: Calculate combinations for Ms. Curren's class

Using the combination formula with $n = 23$ (number of students in Ms. Curren's class) and $r = 6$ (number of slots for Ms. Curren's class). So $C(23,6)=\frac{23!}{6!(23 - 6)!}=\frac{23!}{6!×17!}=\frac{23\times22\times21\times20\times19\times18}{6\times5\times4\times3\times2\times1}=100947$.

Step3: Find total number of possible groups

By the multiplication - principle, the total number of possible groups of students to fill the 11 slots is the product of the number of ways to choose students from Mr. Bentley's class and Ms. Curren's class. So the number of possible groups is $C(21,5)\times C(23,6)=20349\times100947$.

Answer:

$20,349\cdot100,947$