10. Reginald wants to buy a new collar for each of his 3 dogs. The collars come in a choice of 7 different colors. Step 1. How many selections of collars for the 3 dogs are possible if repetitions of colors are allowed? Answer: Step 2. How many selections of collars are possible if repetitions of colors are not allowed? nswer:
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
![### Educational Resources on Combinatorics and Probability
#### Problem 18
**Question:** A value meal package at Ron's Subs consists of a drink, a sandwich, and a bag of chips. There are 5 types of drinks to choose from, 5 types of sandwiches, and 3 types of chips. How many different value meal packages are possible?
**Answer:** 75
*Explanation:* The number of different value meal combinations can be calculated by multiplying the number of choices for each component:
\[ \text{Total packs} = 5 \, (\text{drinks}) \times 5 \, (\text{sandwiches}) \times 3 \, (\text{chips}) = 75 \]
#### Problem 19
**Question:** Evaluate the following expression: \( 12! \)
**Answer:** 479,001,600
*Explanation:* The notation \( 12! \) represents the factorial of 12, which is the product of all positive integers up to 12:
\[ 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 479,001,600 \]
#### Problem 20
**Question:** Reginald wants to buy a new collar for each of his 3 dogs. The collars come in a choice of 7 different colors.
**Step 1:** How many selections of collars for the 3 dogs are possible if repetitions of colors are allowed?
**Answer:** ____
*Explanation:* Assuming each dog can have any of the 7 different colors, and the same color can be used more than once, the number of possible combinations is:
\[ 7 \, (\text{colors for Dog 1}) \times 7 \, (\text{colors for Dog 2}) \times 7 \, (\text{colors for Dog 3}) = 7^3 = 343 \]
**Step 2:** How many selections of collars are possible if repetitions of colors are not allowed?
**Answer:** ____
*Explanation:* If each dog must have a different collar color, the number of possible selections can be calculated using permutations:
\[ 7 \, (\text{choices for Dog 1}) \times 6 \, (\text{choices](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc019303-f71f-44c9-a166-98b7cccf70ca%2F2bb850ff-ca70-44ea-8db1-c6cf967d1d4c%2Ffxa3i8o_processed.jpeg&w=3840&q=75)
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