Answer :
To determine how many different groups of five jurors can be selected from a panel of eleven people, we use the concept of combinations. This concept is applicable because the order in which we select the jurors does not matter, only the group of five people itself.
The formula for calculating combinations, also known as "n choose k," is given by:
[tex]\[ C(n, k) = \frac{n!}{k!(n-k)!} \][/tex]
where:
- [tex]\( n \)[/tex] is the total number of items to choose from (in this case, 11 people).
- [tex]\( k \)[/tex] is the number of items to choose (in this case, 5 jurors).
Plugging in the values:
[tex]\[ C(11, 5) = \frac{11!}{5!(11-5)!} \][/tex]
[tex]\[ C(11, 5) = \frac{11!}{5! \cdot 6!} \][/tex]
Next, factorial values need to be considered:
- [tex]\( 11! \)[/tex] (11 factorial) is the product of all positive integers up to 11.
- [tex]\( 5! \)[/tex] (5 factorial) is the product of all positive integers up to 5.
- [tex]\( 6! \)[/tex] (6 factorial) is the product of all positive integers up to 6.
By performing the necessary calculations:
[tex]\[ \frac{11!}{5! \cdot 6!} = 462 \][/tex]
Thus, there are 462 different ways to choose 5 jurors from a panel of 11 people.
Therefore, the correct answer is:
(B) 462
The formula for calculating combinations, also known as "n choose k," is given by:
[tex]\[ C(n, k) = \frac{n!}{k!(n-k)!} \][/tex]
where:
- [tex]\( n \)[/tex] is the total number of items to choose from (in this case, 11 people).
- [tex]\( k \)[/tex] is the number of items to choose (in this case, 5 jurors).
Plugging in the values:
[tex]\[ C(11, 5) = \frac{11!}{5!(11-5)!} \][/tex]
[tex]\[ C(11, 5) = \frac{11!}{5! \cdot 6!} \][/tex]
Next, factorial values need to be considered:
- [tex]\( 11! \)[/tex] (11 factorial) is the product of all positive integers up to 11.
- [tex]\( 5! \)[/tex] (5 factorial) is the product of all positive integers up to 5.
- [tex]\( 6! \)[/tex] (6 factorial) is the product of all positive integers up to 6.
By performing the necessary calculations:
[tex]\[ \frac{11!}{5! \cdot 6!} = 462 \][/tex]
Thus, there are 462 different ways to choose 5 jurors from a panel of 11 people.
Therefore, the correct answer is:
(B) 462
