Question

How many different ways are there to choose a dozen donuts from the 4 varities at a donut shop so that you get atleast one donut of every variety g

Answer 1

There are 330 different ways for choosing a dozen donuts from the 4 varieties at a donut shop. (at least one donut of every variety must be selected)

How to calculate the number of ways to select items?

There are 'r' items from 'n' different varieties (repetition allowed). Then, the number of ways to select items is given by

The number of ways =  $_{n+r-1}C_r$

Where the combination $_{n}C_r$ is calculated as

$_{n}C_r=\frac{n!}{r!(n-r)!}$

Calculation:

It is given that there are 4 varieties of donuts in a shop. I.e., n = 4

Number of donuts to be selected r = 12 (one dozen)

And also given that at least one donut of every variety has to be selected.

Since there are 4 varieties, at least one from each of these means the count is 4.

So, the remaining number of donuts to be selected is 12 - 4 = 8.

So, r becomes 8 i.e., r = 8

On substituting,

the number of ways of selecting the remaining 8 donuts = $_{4+8-1}C_4$

⇒ $_{11}C_4$

⇒ $\frac{11!}{4!(11-4)!}$

⇒ $\frac{11!}{(4!)(7!)}$

⇒ 330

Therefore, there are 330 different ways for choosing a dozen donuts from the 4 varieties at a donut shop.

Learn more about combinations here:

https://brainly.com/question/11732255

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