Respuesta :
General category: Mathematics
Sub-category: Probability
Topic: counting techniques
Introduction:
The combination formula C(n,r) shows us the number of ways of picking r unordered outcomes from n possibilities. For n ≥ r ≥ 0, C(n,r) is given by the following formula:
[tex]C(n,r)=\frac{n!}{r!(n\text{ -r})!}[/tex]Explanation:We can use combinations and fundamental counting principles to answer this question.
Let us denote by "a", the number of selection of appetizers. This number can be calculated as follows:
[tex]C(3,2)=\frac{3!}{2!(3\text{ -2})!}=3[/tex]Now, let us denote by "b", the number of selection of main courses. This number can be calculated as follows:
[tex]C(7,3)=\frac{7!}{3!(7\text{ -3})!}=35[/tex]Finally, let us denote by "c", the number of selection of desserts. This number can be calculated as follows:
[tex]C(4,3)=\frac{4!}{3!(4\text{ -3})!}=4[/tex]Now, applying the multiplication principle we get the desired number:
[tex]C(3,2)\cdot C(7,3)\cdot C(4,3)=3\cdot35\cdot4\text{ = 420}[/tex]We can conclude that the correct answer is:
