A cafeteria offers 5 types of sandwiches. Moreover, for each type of sandwich, a customer can choose one of 4 breads and opt for either small or large sized sandwich. Optionally, the customer may also add up to 2 out of 6 available sauces. The number of different ways in which an order can be placed for a sandwich, is

The correct answer is 880.

To determine the total number of ways an order can be placed for a sandwich, we need to calculate the number of choices available for each component of the order and then multiply them together using the fundamental counting principle.

Let's break down the choices step-by-step:

Step 1: Choose the sandwich type
The cafeteria offers 5 different types of sandwiches. So, there are 5 ways to choose the sandwich type.

Step 2: Choose the bread
For each sandwich type, a customer can select from 4 different breads. So, there are 4 ways to choose the bread.

Step 3: Choose the size
A sandwich can be either small or large. This gives us 2 ways to choose the size.

Step 4: Choose the sauces
The customer can optionally add "up to 2" out of 6 available sauces. This means the customer can choose exactly 0 sauces, 1 sauce, or 2 sauces. We use combinations (ⁿC_r) to find the number of ways for each scenario:

Choosing 0 sauces: ⁶C₀ = 1 way.

Choosing 1 sauce: ⁶C₁ = 6 ways.

Choosing 2 sauces: ⁶C₂ = (6 × 5) / (2 × 1) = 15 ways.

The total number of ways to choose the sauces is the sum of these possibilities:

$1+6+15=22$

So, there are 22 ways to select the sauces.

Step 5: Calculate the total number of ways
Finally, we multiply the number of choices for each component to find the overall number of different orders that can be placed:

Total ways = (Choices for sandwich type) × (Choices for bread) × (Choices for size) × (Choices for sauces)

$\text{Total ways}=5\times 4\times 2\times 22$

$\text{Total ways}=40\times 22=880$

Therefore, there are 880 different ways to order a sandwich.