Question

In a regional spelling bee, the 8 finalists consist of 3 boys and 5 girls. Find the number of sample points in the sample space S for the number of possible orders at the conclusion of the contest for (a) all 8 finalists; (b) the first 3 positions.

a). 8!,3!
b). 8!,5!
c) 3!,8!
d) 5!,8!

Answer 1

The correct representation for the number of sample points in the sample space
`\(S\)` for the number of possible orders at the conclusion of the contest is (a)
`\(8!\)` for all 8 finalists and (b)
`\(8!,5!\)` for the first 3 positions.

Step-by-step explanation:

(a) For the number of possible orders for all 8 finalists, we consider the total number of arrangements, which is given by
`\(8!\)` (8 factorial). This accounts for all permutations of the 8 finalists regardless of their gender. The correct option is (a)
`\(8!\).`

(b) To find the number of possible orders for the first 3 positions, we consider the arrangement of the first 3 finalists. Since there are 3 boys and 5 girls, the number of ways to arrange the first 3 positions is
`\(8!/(8-3)! = 8!/5!\).` This represents the permutation of 3 boys out of the total 8 finalists for the first 3 positions. The correct option is (b)
`\(8!,5!\).`

Understanding the concept of permutations is essential in combinatorics to calculate the number of ways elements can be arranged. The factorial notation, denoted by
`\(n!\)`, represents the product of all positive integers up to
`\(n\)`. In this context,
`\(8!\)` represents the total number of ways to arrange all 8 finalists, and
`\(8!,5!\)` represents the number of ways to arrange the first 3 positions with consideration for the gender distribution.