You have spent a very long time training your dog to open the refrigerator and bring you a drink when you ask him to. Since your pet can’t read, his selection of the specific type of beverage he brings is completely random – although you notice that, for reasons beyond your understanding, and aside from whatever else he brings, he always brings you 2 (and only 2) grapefruit juices on any given day. One particular day, you ask your pet to bring you 4 drinks. You know that there are 6 bottles of orange juice, 4 bottles of grapefruit juice, and 3 bottles of apple juice in your fridge. Of all the possible combinations of types of drinks that your dog could select, what proportion of those combinations will contain the requisite 2 bottles of grapefruit juice that you know he will inevitably bring?

To determine the proportion of combinations that will contain the requisite 2 bottles of grapefruit juice, we can use the concept of combinations.

First, let's calculate the total number of possible combinations that your dog can bring. Since your dog will bring you 4 drinks and there are a total of 13 bottles in the fridge (6 orange juice, 4 grapefruit juice, and 3 apple juice), we can use the combination formula:

nCr = n!/(r!(n-r)!)

In this case, n (total number of bottles) is 13 and r (number of drinks your dog will bring) is 4. Let's calculate:

13C4 = 13!/(4!(13-4)!) = 715

So, there are 715 possible combinations of drinks that your dog could bring.

Now, let's consider the combinations that contain the requisite 2 bottles of grapefruit juice. Since your dog always brings exactly 2 grapefruit juices, we need to select the remaining 2 drinks from the remaining bottles.

There are 4 grapefruit juice bottles, so we can calculate the combination of selecting 2 grapefruit juices:

4C2 = 4!/(2!(4-2)!) = 6

Next, we need to select the remaining 2 drinks from the remaining bottles (6 orange juice and 3 apple juice). Therefore, the combination becomes:

(6+3)C2 = 9!/(2!(9-2)!) = 36

Therefore, there are 36 combinations that contain the requisite 2 bottles of grapefruit juice.

To calculate the proportion of combinations, we divide the number of combinations with 2 grapefruit juice bottles (36) by the total number of possible combinations (715):

Proportion = 36/715 ≈ 0.0503 (rounded to four decimal places)

So, approximately 0.0503 or 5.03% of all possible combinations will contain the requisite 2 bottles of grapefruit juice that your dog will inevitably bring.