Question

A closet has 5 pairs of different types of shoes. The number of ways in which 4 shoes can be drawn from it such that there will be no complete pair is (1) 200 (2) 160 (3) 40 (4) 80

Short answer

80

Step-by-step solution

- Understand the Problem

A closet contains 5 pairs of different types of shoes (10 shoes total). We need to select 4 shoes in such a way that no complete pair is included in the selection.

- Calculate Total Possible Ways

To find this, calculate the total number of ways to pick any 4 shoes out of 10. This is given by the combination formula \(\binom{10}{4}\): \[\binom{10}{4} = \frac{10!}{4!(10-4)!} = 210\]

- Calculate Invalid Combinations

Next, calculate the number of ways to draw 4 shoes such that at least one complete pair is included. For each pair included, the remaining two shoes are selected from the remaining 8 shoes. This uses \(\binom{5}{1} \times \binom{8}{2}\): \[\binom{5}{1} \times \binom{8}{2} = 5 \times 28 = 140\]

- Calculate Valid Combinations

Finally, subtract the number of invalid combinations (where at least one complete pair is included) from the total combinations to get the number of ways 4 shoes can be drawn such that no pair is included: \[210 - 140 = 70\]

- Correct Calculation

There appears to be an inconsistency, so check the correct logical reasoning and problem-solving approach: Choose 4 shoes out of 5 pairs such that no pairs are completed. First, select 4 pairs from 5 pairs, and then choose one shoe from each selected pair. \[\binom{5}{4} \times 2^4 = 5 \times 16 = 80\]

Key concepts

Combinatorial Analysis

Combinatorial analysis involves the study of counting, arrangement, and combination of elements in sets. It's a fundamental area in mathematics widely used to solve selection problems. Let's break it down: the given problem involves selecting a subset from a larger set under certain constraints.
The techniques used here include combinations, where order doesn't matter, and permutations, where order does matter. For instance, selecting 4 shoes from 10 different shoes falls under combinations because the order in which you select them doesn’t matter.
In this problem, exploring both valid and invalid combinations helps ensure a complete understanding. This often involves a careful analysis of different scenarios to ensure the solution covers all possible cases.

Permutations and Combinations

Understanding permutations and combinations is crucial for solving selection problems. Here are the basics:

• Permutations: The arrangement of objects where order matters. For example, there are 6 ways to arrange the letters A, B, C (ABC, ACB, BAC, BCA, CAB, CBA).
• Combinations: The selection of objects where order does not matter. For example, choosing 2 out of 3 letters A, B, C has 3 combinations (AB, AC, BC).

In the given exercise, combinations are used to determine the ways to choose 4 shoes from 10 without considering order. \[ \binom{10}{4} = \frac{10!}{4!(10-4)!} \] Simplifying, we find 210 ways to pick any 4 shoes.
However, we also account for invalid combos where at least one complete pair is chosen using \[ \binom{5}{1} \times \binom{8}{2} \] resulting in 140 ways. Finally, valid combinations are tallied by subtracting these invalid choices.

Probability in Selection Scenarios

Probability in selection scenarios often hinges on combinatorial methods. It involves calculating the likelihood of specific outcomes from a set of possibilities. Consider our shoe-selection example. We begin by finding the total number of ways to draw 4 shoes: 210.
Next, we find the number of ways to draw invalid combinations involving complete pairs: 140. Subtracting these invalid combinations from the total gives valid ways to select 4 shoes without a complete pair: 70.
Reassessing the problem, we see we need to choose 4 shoes such that no complete pair is included directly by: 1. Selecting 4 out of 5 pairs (\binom{5}{4}). 2. Choosing one shoe from each selected pair (2^4).
Combining these calculations gives \[ \binom{5}{4} \times 2^4 \] yielding 80 valid ways. Understanding these probability scenarios enhances clarity on possible outcomes.