Nemo has 5 swimsuits, 2 pairs of flippers, and 3 pairs of goggles, which can be worn in any combination. He needs to choose a combination of swimwear to wear to a family pool party. How many different combinations consisting of 1 swimsuit, 1 pair of flippers, and 1 pair of goggles are possible for Nemo to wear to the pool party?
The Correct Answer and Explanation is:
To find how many different combinations Nemo can wear to the pool party, we use the fundamental counting principle. This principle states that if there are m ways to do one thing, n ways to do another, and p ways to do a third, then the total number of combinations is the product of the individual choices.
Nemo has the following:
- 5 swimsuits
- 2 pairs of flippers
- 3 pairs of goggles
Each full outfit for the pool party must include 1 swimsuit, 1 pair of flippers, and 1 pair of goggles.
Step-by-step Calculation:
- Choices for swimsuits: 5
- Choices for flippers: 2
- Choices for goggles: 3
Now multiply the number of choices: 5×2×3=305 \times 2 \times 3 = 305×2×3=30
Final Answer:
30 different combinations
Explanation
This is a classic example of applying the fundamental counting principle in combinatorics. The principle helps in counting the total number of possible outcomes when a task is broken down into a sequence of independent choices.
In this case, Nemo is preparing for a pool party and needs to select one item from each category of swimwear: swimsuits, flippers, and goggles. Since each category is independent of the others, the number of total combinations is found by multiplying the number of options in each category.
He has 5 swimsuits, which means he can begin by choosing any one of these five. For each of these swimsuit choices, he can pair it with any of the 2 flippers. That gives us 5 × 2 = 10 combinations so far.
But the outfit is not complete yet. For each of these 10 swimsuit-flipper combinations, he can choose from 3 pairs of goggles. So, we multiply 10 by 3, resulting in 30.
These combinations represent all the unique ways Nemo can dress for the pool party without repeating any item more than once in a single outfit. It’s important to note that the order of wearing the items does not matter; what matters is the combination of one from each group. Thus, Nemo has 30 different outfit combinations available.
