Final answer:
The probability that the purchase will consist of at least one of each item (dog treat, toy, and piece of clothing) is calculated by multiplying the number of options in each category and dividing by the total number of outcomes for any three items, giving a probability of approximately 27.27%.
Step-by-step explanation:
To find the probability that the purchase will consist of at least one of each item (one dog treat, one toy, and one piece of clothing), we must consider all possible combinations of these items.
The total number of outcomes for buying any three items without any restriction can be found by adding the individual possibilities for each category and then choosing three items:
(3 dog treats + 4 toys + 5 pieces of clothing) choose 3 = 12 choose 3.
To compute 12 choose 3, we use the combination formula which is C(n, k) = n! / (k!(n - k)!), where 'n' is the total number of items and 'k' is the number of items to choose.
So, 12 choose 3 = 12! / (3!(12 - 3)!)
= 220.
To find the outcomes that have at least one of each item, we need to calculate the product of the individual possibilities from each category, since we are choosing one from each:
3 dog treats * 4 toys * 5 pieces of clothing = 60.
Hence, the total number of outcomes with at least one of each item is 60.
The correct probability is therefore 60 / 220 = 0.2727 or about 27.27%.