Problem 59 - A Complete Resource Book in Mathematics for JEE Main

Question

There are stalls for 10 animals in a ship. The number of ways the shipload can be made if there are cows, calves and horses to be transported, animals of each kind being not less than 10 , is (A) 59049 (B) 49049 (C) 69049 (D) None of these

Step-by-Step Solution

Answer

The number of ways is 66, which doesn't match any given option, so (D) None of these.

Step 1: Understanding the Problem

We are given 10 stalls in a ship and need to fill them using 10 or more animals of three types: cows, calves, and horses. We need to find how many ways to fill these stalls with the constraint that the total number of animals should be 10.

Step 2: Using the Multinomial Coefficient Formula

Since each type of animal can fill one or more stalls, this is a combinatorial problem. We use the stars and bars theory, which can be applied using the formula for distributing n identical items into k distinct groups: \( \binom{n+k-1}{k-1} \).

Step 3: Applying the Formula

Applying the above formula with \( n=10 \) stalls and \( k=3 \) types of animals, we calculate: \( \binom{10+3-1}{3-1} = \binom{12}{2} \).

Step 4: Calculating the Binomial Coefficient

Calculating \( \binom{12}{2} \): \( \frac{12 \cdot 11}{2 \cdot 1} = 66 \). Therefore, there are 66 ways to fill the stalls with the given conditions.

Key Concepts

Stars and Bars Theory

The stars and bars theory is a fascinating concept in combinatorics that helps solve problems related to distributing identical items into distinct groups. Picture each item as a "star," and the divisions between different groups as "bars." This mathematical theory works by determining how to arrange these stars and bars linearly, with the number of bars representing the number of divisions needed.

For instance, if you are distributing 10 identical stars (animals) into 3 groups (cows, calves, horses), you need 2 bars to create 3 partitions. Stars and bars theory effectively transforms the problem into finding different ways to place these bars among the stars. This visual or conceptual representation allows you to better understand the distribution challenges when dealing with identical objects.

It's important to use stars and bars theory when each group can receive zero or more items. However, our exercise requires that each group has at least one star, but the principle still applies with adjusted calculations using the binomial coefficient.

Multinomial Coefficient

The multinomial coefficient is a generalization of the binomial coefficient and is an essential tool in combinatorics. It is used when we need to distribute objects into more than two groups. In formulaic terms, it is expressed as:\[ \binom{n}{k_1, k_2, ..., k_m} = \frac{n!}{k_1! \, k_2! \, ... \, k_m!} \]Here, \( n \) represents the total number of items to distribute, while \( k_1, k_2, ..., k_m \) are the amounts going into each group. The factorial symbol (!) represents the product of all positive integers up to that number.

In the given exercise, the multinomial coefficient could be applied if there were specific numbers of each type of animal required. However, since any combination is acceptable as long as the total is 10, we use a version of this idea applied through stars and bars theory. The real power of multinomial coefficients lies in their ability to handle distributions where the group sizes are predetermined and fixed.

Binomial Coefficient

The binomial coefficient is foundational in combinatorics and is used to determine the number of ways to choose \( k \) items from a set of \( n \) items without regard for the order of selection. It is denoted as \( \binom{n}{k} \), and can be calculated through the formula:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]In our current problem, we were tasked with placing 10 animals into 3 stalls. Using the stars and bars approach, the binomial coefficient provides the exact count of ways to do so by rearranging 10 stars and 2 bars. In this situation, we compute \( \binom{12}{2} \), fundamentally reflecting the number of ways to choose 2 dividers within a sequence of 10 stars (animals) plus 2 bars (group dividers).

The usefulness of the binomial coefficient extends far beyond counting distributions; it is essential for understanding probabilities, algebra, and more advanced mathematical formulations in fields like statistics and computer science. Keep in mind, solving such exercises strengthens your combinatorial reasoning and problem-solving skills, especially in applying the right coefficient to find your solution.