Respuesta :
Answer: The total number of ways are 792 ways.
Explanation:
The total number of spots available is 7.
The total number of eligible candidate is 12.
To find the number of combinations we have a combination formula.
[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]
Where n is total number of of objects and r is the number of selected objects.
Since we have to select 7 spots from 12 candidates, therefore the total number of ways is calculated as shown below,
[tex]^{12}C_7=\frac{12!}{7!(12-7)!}[/tex]
[tex]^{12}C_7=\frac{12\times 11\times 10\times 9\times 8\times7!}{7!5!}[/tex]
[tex]^{12}C_7=\frac{12\times 11\times 10\times 9\times 8}{5\times 4\times 3\times 2\times 1}[/tex]
[tex]^{12}C_7=792[/tex]
Therefore, the total number of ways are 792 ways.
Using the permutation formula, it is found that the members of the cabinet can be appointed in 3,991,680 ways.
In this problem, rank matters, hence the permutation formula is used.
What is the permutation formula?
The number of possible permutations of x elements from a set of n elements is given by:
[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]
In this problem, 7 candidates are chosen from a set of 12, hence:
[tex]P_{(12,7)} = \frac{12!}{5!} = 3991680[/tex]
The members of the cabinet can be appointed in 3,991,680 ways.
More can be learned about the permutation formula at https://brainly.com/question/25925367
