Question

How many different ways are there to choose a dozen donuts from the 21 varieties at a donut shop?

Answer 1

To find the number of ways to choose a dozen donuts from 21 varieties, we use the formula for combinations C(21, 12). This represents a combination without repetition, and requires calculating 21! divided by the product of 12! and the factorial of (21-12), yielding the final solution.

Step-by-step explanation:

To determine how many different ways there are to choose a dozen donuts from the 21 varieties at a donut shop, we need to use the concept of combinations in combinatorial mathematics. This kind of problem is a typical example of a combination without repetition, where the order of selection does not matter and each selection is unique.

In this case, the formula we use is the binomial coefficient, which is usually represented as nCr or C(n, r), where n is the total number of items to choose from (in this case, 21 donut varieties), and r is the number of items to choose (here, a dozen or 12 donuts).

Calculating Combinations

1. First, we list our n and r values: n = 21, r = 12.
2. Next, we use the combination formula: C(n, r) = n! / (r!(n-r)!).
3. Calculating this, we get C(21, 12) = 21! / (12! * (21-12)!).

Using a calculator or combinatorial function in a programming language or mathematical software, we would find the precise number of ways to make our selection of donuts.

Answer 2

352716

Step-by-step explanation:

Here all we have to do is

counting the number of possible combinations of 10 elements that we can form out of a set of 21 elements.

then

The number of ways = 21C10 = 352716

Note :

`21C10=C^(10)_(21)=(21!)/(10!* \left( 21-10\right) !)`