Question

Suppose that a restaurant offers a homestyle vegetable dinner with a choice of four side dishes, and suppose that there are 26 side dishes to choose from.

In how many ways can one order the dinner, if the side dishes are brought out simultaneously (and thus, their order is irrelevant), and a side dish cannot be ordered more than once?

Answer 1

The number of ways to order four side dishes from a choice of 26, without repetition and where order does not matter, is calculated using the combination formula. The formula C(26, 4) simplifies to 14,950 ways.

Step-by-step explanation:

To determine the number of ways one can order a homestyle vegetable dinner with a choice of four side dishes from 26 available options, when the order is irrelevant and no dish can be repeated, we use the combination formula. The formula for combinations is C(n, k) = n! / (k!(n-k)!), where n is the total number of options and k is the number selected.

Applying this formula to our scenario, we get C(26, 4) = 26! / (4!(26-4)!), which simplifies to 26! / (4!22!). This represents the number of combinations of 26 items taken 4 at a time without regard to order.

Step-by-step Calculation

1. Calculate 26! (factorial of 26).
2. Calculate 4! (factorial of 4).
3. Calculate 22! (factorial of 22).
4. Divide 26! by the product of 4! and 22!:

C(26, 4) = 26! / (4! * 22!) = (26 * 25 * 24 * 23) / (4 * 3 * 2 * 1)

After simplification, we get:

C(26, 4) = 14,950

Therefore, one can order the homestyle vegetable dinner with four side dishes in 14,950 different ways.