Respuesta :
27 different dessert platters can be offered by restaurant
Solution:
Given that,
A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter
Each dessert platter contains an equal number of kinds of cheese and kinds of fruit
For this problem there are two scenarios:
1 ) one cheese and one fruit
2 ) two cheese and two fruit
For one cheese and one fruit
[tex]6C_1 \times 2C_1[/tex]
Use the combination formula
[tex]C(n, r) = \frac{n !}{r ! (n-r) ! }[/tex]
Where, n is total items and r is the items being chosen at a time
[tex]6C_1 \times 2C_1 = \frac{ 6 ! }{ 1 ! (6 - 1) ! } \times \frac{ 2 ! }{ 1 ! ( 2 -1 ) ! }\\\\6C_1 \times 2C_1 = \frac{ 6 ! }{ 1 ! 5 ! } \times \frac{ 2 ! }{ 1 ! 1 ! }\\\\6C_1 \times 2C_1 = \frac{ 6 \times 5 \times 4 \times 3 \times 2 \times 1}{ 5 \times 4 \times 3 \times 2 \times 1} \times 2 \times 1\\\\6C_1 \times 2C_1 = 6 \times 2 = 12[/tex]
For 2 cheese and 2 fruits
[tex]6C_2 \times 2C_2[/tex]
[tex]6C_2 \times 2C_2 = \frac{ 6 ! }{ 2 ! ( 6 - 2) ! } \times 2C_2\\\\We\ know\ that\ 2C_2 = 1\\\\6C_2 \times 2C_2 = \frac{ 6 ! }{ 2 ! ( 6 - 2) ! } \times 1\\\\6C_2 \times 2C_2 = \frac{ 6 \times 5 \times 4 \times 3 \times 2 \times 1 }{2 \times 1 \times 4 \times 3 \times 2 \times 1 }\\\\6C_2 \times 2C_2 = 15[/tex]
So, total ways = 12 + 15 = 27
Thus, 27 different dessert platters can be offered by restaurant
