Question

Three boys and two girls stand in a queue. The probability, that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is (A) \(\frac{1}{2}\) (B) \(\frac{1}{3}\) (C) \(\frac{2}{3}\) (D) \(\frac{3}{4}\)

Short answer

The probability is \frac{3}{20}.

Step-by-step solution

Determine total arrangements

The total number of ways to arrange the three boys (B) and two girls (G) in a queue without any restrictions is given by the permutation formula for the 5 individuals, which is 5 factorial. This is calculated as 5! = 5 × 4 × 3 × 2 × 1 = 120 different arrangements.

Calculate favorable arrangements

To find the number of favorable arrangements where each girl has at least one more boy in front of her than the number of girls, we need to consider possible positions for girls. The first girl must be in one of the last four positions (since at least one boy needs to be ahead), and the second girl must be behind the first girl with at least one boy in between.List the favorable cases considering the positions of the girls (G):B G B G B,B B G B G,B B B G G.There are no other possibilities that satisfy the condition. Let's count the arrangements for each case:- For B G B G B: There are 3! ways to arrange the boys and 2! ways to arrange the girls, resulting in 6 arrangements.- For B B G B G: There are also 3! ways to arrange the boys and 2! ways to arrange the girls, giving another 6 arrangements.- For B B B G G: There are 3! = 6 ways to arrange the boys and the girls are fixed, resulting in 6 arrangements.The total number of favorable arrangements is the sum of the above which is 6 + 6 + 6 = 18.

Calculate the probability

The probability of the favorable event is the number of favorable arrangements divided by the total number of arrangements. That is, Probability = Number of Favorable Arrangements / Total Arrangements = 18 / 120. Simplify the fraction by dividing both numerator and denominator by 6. The probability is thus 3 / 20.

Key concepts

Permutations and Combinations

Understanding permutations and combinations is crucial for solving many mathematics problems, especially when it comes to arranging objects or people. In permutations, the order matters, like in a queue or a ranked list. Combinations, on the other hand, are about grouping without regard to the order.

For instance, if we wanted to find the number of ways to arrange three boys and two girls in a queue, we'd use permutations because the order in which they stand matters. This is calculated using the factorial function, denoted as 'n!'. The factorial represents the product of all positive integers up to a number n. So for our five individuals, the permutations would be 5! which equals 120 different arrangements.

However, if we were interested in knowing how many ways we can form teams of, say, two people from a group, where the order doesn't matter, we'd use combinations. To calculate this, we would use the formula Cr = n! / (r!(n-r)!), where 'n' is the total number of items, and 'r' is the number of items being chosen. Permutations and combinations both play a pivotal role in the field of combinatorics, an area of mathematics concerning the counting, arrangement, and combination of objects.

JEE Advanced Mathematics

JEE Advanced Mathematics is known for its challenging and intricate problems, which require a deep understanding of various mathematical concepts, including permutations and combinations, calculus, algebra, and more. Aspiring to clear the Joint Entrance Examination (JEE) Advanced — a prerequisite for admission into many of India's premier engineering institutes like IITs — students must harness their problem-solving skills and apply them efficiently.

The problem we delve into exemplifies a typical JEE Advanced problem that tests understanding of both, probability as well as combinatorics. It is essential for students to not only learn the core concepts but to also develop the ability to apply these concepts in unconventional ways to solve complex problems. Emphasizing conceptual clarity and practice with multiple problem types is crucial as students prepare for such competitive exams.

Probability Concepts

Probability is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates that the event will not occur, and 1 indicates that the event will definitely occur. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

In the queue problem, we calculate the probability of a specific arrangement of boys and girls by determining the number of favorable arrangements that satisfy given conditions and dividing it by the total number of possible arrangements. To do so, we have to be mindful of the restrictions placed by the problem — an excellent example of conditional probability, where the probability of an event depends on one or more other events.

After calculating both the favorable arrangements and the total possible arrangements, students should be comfortable with simplifying their answers by finding common factors to reduce fractions to their simplest form, as seen in the step-by-step solution offered. Grasping such probability concepts and their applications are immensely helpful in solving problems across various topics in mathematics and statistics.