Question

Amanda goes to the toy store to buy 1 ball and 3 different board games. If the toy store is stocked with 3 types of balls and 6 types of board games, how many different selections of the 4 items can Amanda make? a. 9 b. 12 c. 14 d. 15 e. 60

Short answer

60 selections (option e)

Step-by-step solution

- Choose the Ball

Amanda needs to choose 1 ball from 3 types of balls. Therefore, the number of ways to choose the ball is 3.

- Choose the Board Games

Amanda needs to choose 3 different board games from 6 types of board games. The number of ways to do this can be calculated using combinations: \[\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20\]

- Calculate Total Selections

To find the total number of different selections of the 4 items (1 ball and 3 different board games), multiply the number of ways to choose the ball by the number of ways to choose the board games: \[3 \times 20 = 60\]

- Verify Answer

The total different selections of 4 items Amanda can make is 60, which corresponds to option e.

Key concepts

mathematical combinations

In combinatorics, mathematical combinations are used to determine how many ways we can choose a subset from a larger set.
The order in which the items are selected does not matter.
The formula for combinations is given by \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(n\) is the total number of items, and \(r\) is the number of items to choose.
In our problem, Amanda needs to choose 3 board games out of 6, represented as \(\binom{6}{3}\). The factorial notation \(!\) means multiplying a sequence of descending natural numbers.
For instance, \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1\).
By applying the formula, we find \[\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20\], which means there are 20 ways to choose 3 board games.

problem-solving skills

Problem-solving skills are crucial in mathematics and everyday life.
They involve understanding the problem, devising a plan, carrying out the plan, and then reviewing the solution.
In Amanda's toy store problem, identifying the need to choose both a ball and board games is the first step.
Breaking down the selection into smaller parts like choosing the ball first, and then the board games helps simplify the problem.
Multiplying these individual results gives the final count of possible selections.
Regular practice with similar problems can enhance your problem-solving skills by familiarizing you with different strategies and approaches.

GMAT preparation

For GMAT preparation, understanding combinatorics can be extremely helpful.
The GMAT often includes questions where you need to calculate combinations and permutations.
Combining this knowledge with quick mental calculations can improve your performance on the test.
Practicing with a variety of problems, like the one involving Amanda's toy store, ensures that you become comfortable with these types of questions.
Additionally, being able to identify keywords in the problem statement will help you determine whether to use combinations or permutations, which is crucial for efficient problem-solving during the test.

educational mathematics

Educational mathematics aims to build a strong foundation in various mathematical concepts including algebra, geometry, and combinatorics.
Exercises like Amanda's toy store selection help students understand the application of combinations in real-world scenarios.
By working through step-by-step solutions, students can better grasp complex ideas and techniques.
It's important to develop both conceptual understanding and procedural fluency.
Educators often use such problems to highlight the relevance of mathematical theories and formulas in practical decision-making processes.
As students progress, they learn not only to solve these problems but also to apply them in different contexts, thereby enhancing their overall mathematical literacy.