Problem 2 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Word UNIVERSITY is arranged randomly. Then the probability that both \(I\) does not come together is: [UPSEAT-2001] |(a) \(3 / 5\) (b) \(2 / 5\) (c) \(4 / 5\) (d) \(1 / 5\) Solution (c) Total number of ways \(=\frac{10 !}{2 !}\) Favourable number of ways for \(I\) come together is \(9 !\) Thus, probability that \(I\) come together $$ =\frac{9 ! \times 2 !}{10 !}=\frac{2}{10}=\frac{1}{5} $$

Step-by-Step Solution

Answer

The probability that I does not come together is \( \frac{4}{5} \).

Step 1: Calculate Total Arrangements

First, we need to calculate the total number of ways to arrange the letters in the word "UNIVERSITY". This word has 10 letters, with I repeated twice. The formula to calculate permutations with repeated elements is \( \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \), where \( n \) is the number of letters and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters. So, the total number of arrangements is \( \frac{10!}{2!} \).

Step 2: Calculate Arrangements with I Together

To find the number of ways where the two I's come together, we consider the two I's as a single entity. This effectively reduces the problem to arranging 9 entities (8 distinct letters + 1 group of I's). The number of arrangements is \( 9! \). However, since the two I's within the group can be arranged in \( 2! \) ways, the total is \( 9! \times 2! \).

Step 3: Determine Probability of I Together

The probability that the I's come together is given by the ratio of favorable cases to total cases. Therefore, it is \( \frac{9! \times 2!}{\frac{10!}{2!}} = \frac{2}{10} = \frac{1}{5} \).

Step 4: Calculate Probability of I Not Together

To find the probability that the I's do not come together, we subtract the probability of them coming together from 1 (since these two events are complementary). Hence, \( 1 - \frac{1}{5} = \frac{4}{5} \).

Key Concepts

Understanding Probability Concepts

Probability is an essential concept in mathematics that helps us measure the chance or likelihood of an event occurring. It ranges from 0 to 1, where 0 indicates impossibility, and 1 indicates certainty. In the context of permutations, probability allows us to determine how likely a particular arrangement occurs among all possible configurations.
When dealing with permutations, the probability formula is often expressed as the ratio of the number of favorable outcomes to the total number of possible outcomes. For example, when determining the probability that two 'I's come together in the word "UNIVERSITY", we first calculate the total arrangements and then the specific arrangements where 'I's are grouped together.
Thus, the probability of the 'I's not coming together is obtained by recognizing that these are either-or events, meaning they are complementary. So, one can subtract the probability of the event that interests us from 1. This is an important technique in solving probability problems effectively.

Factorial Calculation Simplified

Factorial calculation is a fundamental operation in permutations and probability. Denoted by an exclamation mark, for instance, 10!, it represents the product of all positive integers up to that number. Therefore, 10! is the product of all integers from 1 to 10.
Factorials are especially powerful in problems involving permutations where the sequence or arrangement of items matters. In our permutation example involving "UNIVERSITY", which has 10 letters where 'I' repeats twice, the value is adjusted using factorial division: \( \frac{10!}{2!} \). This division corrects for the repeated element by reducing redundant permutations.
Calculating factorials may seem daunting, but they hold the key to understanding how many different ways items can be arranged, forming the backbone of both probability and combinatorial analyses.

Combinatorial Analysis in Practice

Combinatorial analysis is the branch of mathematics dealing with counting and arrangements, crucial for solving permutation problems. It encompasses principles that allow us to explore different ways items can be selected and arranged.
In the "UNIVERSITY" problem, we use combinatorial analysis to handle repeated elements and specific conditions, such as 'I's coming together. By treating these 'I's as one unit, we simplify the problem significantly, allowing the calculation of permutations as if one less letter exists.
Combinatorial analysis not only helps in counting possible outcomes but also sets up the framework for calculating probabilities. Recognizing patterns, spotting duplication issues, and leveraging these techniques enables more efficient and accurate outcomes in permutation problems.