Question

Kareem has 4 sweaters, 6 shirts, and 3 pairs of slacks. How many distinct outfits, each consisting of a sweater, a shirt, and a pair of slacks, can Kareem select? F. 13 G. 36 H. 42 J. 216 K. 216

Short answer

Answer: 72

Step-by-step solution

Identify the number of choices for each item

Kareem has 4 choices for sweaters, 6 choices for shirts, and 3 choices for pairs of slacks.

Apply the fundamental counting principle

Using the fundamental counting principle, we need to multiply the number of choices for each item together to find the total number of distinct outfits. This can be written as: Total outfits = Number of sweaters * Number of shirts * Number of slacks

Calculate the total number of distinct outfits

Now, we will plug in the values to calculate the total number of distinct outfits Kareem can select: Total outfits = 4 * 6 * 3 Total outfits = 24 * 3 Total outfits = 72 So Kareem can select 72 distinct outfits. Since 72 is not one of the answer choices, we need to revise our calculations to check for any errors or mistakes in the process. Upon revisiting the problem, it appears that the error is in the answer choices provided in the exercise. The correct answer, 72, is missing from the options (F, G, H, J, K). Please inform the person who created the question about the error, and proceed with the correct answer of 72 distinct outfits.

Key concepts

Permutations

Permutations are at the heart of many combinatorial problems, where the order of arrangement is important. A classic example is arranging letters to form words, where the word 'CAT' is different from 'ACT' because the order of the letters matters. In mathematics, we define a permutation as an ordered arrangement of objects without repetition.

In the context of everyday scenarios like choosing outfits, permutations help us determine the number of possible arrangements where the order of selection is significant. However, in the given exercise with Kareem's sweaters, shirts, and slacks, the order does not matter, thus we don't actually apply permutations here. Instead, we use the simpler fundamental counting principle. Nevertheless, understanding permutations is crucial for problems where positioning is key, such as seating arrangements or assigning ranks.

Combinatorics

Combinatorics is the branch of mathematics that studies counting, combinations, and permutations. It's fundamentally about finding the number of different ways things can be arranged or combined, which is precisely what we're doing when figuring out Kareem's outfit combinations. In this context, combinatorics tells us we need to consider each category (sweaters, shirts, slacks) independently and then use the fundamental counting principle to find the total number of combinations.

The fundamental counting principle, a cornerstone of combinatorics, states that if there are 'n' ways to perform one task, and 'm' ways to perform another, there are n * m ways to perform both tasks sequentially. This principle simplifies the process of finding Kareem's total outfit possibilities to basic multiplication, as demonstrated in the exercise.

ACT Math Preparation

Preparing for the ACT Math test requires familiarity with a wide range of mathematical concepts, including algebra, geometry, and, yes, combinatorics. For students, understanding the fundamental counting principle and permutations is invaluable for quickly solving various types of counting problems that may appear on the test.

For ACT Math preparation specifically, practice problem sets often include examples like the one featuring Kareem's outfit choices to reinforce these concepts. It's essential for students to not only know how to calculate combinations and permutations but also to read and interpret the problems carefully. As shown in the original exercise, misinterpreting the question or making a calculation error could lead to incorrect answer choices. By practicing with a variety of problems, students will become adept at avoiding common pitfalls and will improve their chances of achieving a high score on the ACT Math section.