Question

A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

Answer 1

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

$6C_1 \times 2C_1$

Use the combination formula

$C(n, r) = \frac{n !}{r ! (n-r) ! }$

Where, n is total items and r is the items being chosen at a time

$6C_1 \times 2C_1 = \frac{ 6 ! }{ 1 ! (6 - 1) ! } \times \frac{ 2 ! }{ 1 ! ( 2 -1 ) ! }\\\\6C_1 \times 2C_1 = \frac{ 6 ! }{ 1 ! 5 ! } \times \frac{ 2 ! }{ 1 ! 1 ! }\\\\6C_1 \times 2C_1 = \frac{ 6 \times 5 \times 4 \times 3 \times 2 \times 1}{ 5 \times 4 \times 3 \times 2 \times 1} \times 2 \times 1\\\\6C_1 \times 2C_1 = 6 \times 2 = 12$

For 2 cheese and 2 fruits

$6C_2 \times 2C_2$

$6C_2 \times 2C_2 = \frac{ 6 ! }{ 2 ! ( 6 - 2) ! } \times 2C_2\\\\We\ know\ that\ 2C_2 = 1\\\\6C_2 \times 2C_2 = \frac{ 6 ! }{ 2 ! ( 6 - 2) ! } \times 1\\\\6C_2 \times 2C_2 = \frac{ 6 \times 5 \times 4 \times 3 \times 2 \times 1 }{2 \times 1 \times 4 \times 3 \times 2 \times 1 }\\\\6C_2 \times 2C_2 = 15$

So, total ways = 12 + 15 = 27

Thus, 27 different dessert platters can be offered by restaurant