Problem 65 - Algebra for JEE ADVANCED

Question

A person predicts the outcome of 20 cricket matches of his home team. Each match can result in a either win, loss or tie for the home team. Total number of ways in which he can make the predictions so that exactly 10 predictions are correct is equal to a. \(\quad{ }^{20} C_{10} \times 2^{10}\) b. \(\quad{ }^{20} C_{10} \times 3^{20}\) c. \({ }^{20} C_{10} \times 3^{10}\) d. \({ }^{20} C_{10} \times 2^{20}\)

Step-by-Step Solution

Answer

The correct answer is a: \( ^{20}C_{10} \times 2^{10} \).

Step 1: Understand the Problem

The task is to determine the number of ways a person can correctly predict the outcomes of 10 out of 20 matches where each match has three possible outcomes: win, loss, or tie.

Step 2: Choose Correct Predictions

The number of ways to choose exactly 10 matches out of 20 for which the predictions will be correct is given by the combination formula \( ^{20}C_{10} \). This represents choosing 10 matches from 20 without regard to order.

Step 3: Analyze Remaining Predictions

For the remaining 10 matches, since their predictions are incorrect, there are 2 incorrect options for each match (any outcome other than the correct one). Thus, the number of ways to predict incorrectly for each of these 10 matches is \( 2^{10} \).

Step 4: Calculate Total Number of Ways

By combining the number of ways to choose which 10 matches are predicted correctly with the number of ways to incorrectly predict the remaining matches, the total number of prediction combinations is \( ^{20}C_{10} \times 2^{10} \).

Step 5: Select Correct Answer Option

Compare the calculated expression with the provided answer choices. The correct answer is choice a: \( ^{20}C_{10} \times 2^{10} \).

Key Concepts

Understanding Probability

Probability is a way to quantify the chances of an event occurring. It is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. In this exercise, each cricket match has three possible outcomes: win, loss, or tie. This means for each prediction there are three possible scenarios considered.

Probability helps us assess how likely we are to get a certain result. In this case, predicting the outcomes correctly relies on probability concepts. For instance, if each prediction is made purely by chance, the probability of each being correct is 1 in 3, since only one of the three possible outcomes can be correct.

This concept of probability allows us to formulate expressions and understand the likely outcomes of choosing which matches are predicted correctly.

Binomial Coefficient Essentials

The binomial coefficient, often referred to as "n choose k" and denoted as \( \binom{n}{k} \), calculates the number of ways to select k successes (or selections) from n trials (or items). It does so without regard to the order in which they are selected. This is crucial for understanding the combination formula.

For example, in our problem, \( ^{20}C_{10} \) calculates how many ways 10 matches can be selected correctly out of 20. The formula for the binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \( n! \) (n factorial) is the product of all positive integers up to n.

Using the binomial coefficient helps to break down complex selection problems into manageable steps by focusing on combination rather than permutation, where order matters.

Permutations and Combinations Simplified

Permutations and combinations are fundamental concepts in combinatorics that help in counting arrangements and selections. While permutations focus on ordered arrangements, combinations deal with selections where order does not matter.

For instance, combinations are used in our problem since predicting which matches will be correct is not dependent on any order. We just care about which 10 are picked. This is mathematically expressed with combinations as previously mentioned in the binomial coefficient section.

On the other hand, permutations would apply if we were interested in the order of correct predictions; however, in this scenario, the order is inconsequential. Thus, understanding when to use permutations versus combinations is key in tackling problems, allowing us to compute answers efficiently.