Problem 53 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In a shop there are five types of ice-creams available. A child buys six ice- creams. Statement 1: The number of different ways the child can buy the six ice-creams is \({ }^{10} C_{5}\). Statement 2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging \(6 \mathrm{~A}\) 's and \(4 \mathrm{~B}\) 's in a row. (A) Statement 1 is false, Statement 2 is true (B) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1 (C) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is true, Statement 2 is false

Step-by-Step Solution

Answer

(B) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.

Step 1: Understanding the Problem

The problem asks us to determine the number of ways a child can buy six ice-creams from five available types. We then need to verify the two given statements regarding the solution.

Step 2: Analyzing Statement 1

Statement 1 claims that the number of ways to select six ice-creams from five types is represented as \( ^{10}C_5 \). To analyze this, understand that this is a combination problem where we need to distribute 6 identical items (ice-creams) among 5 distinct categories (types). The problem translates into a 'stars and bars' method which defines the situation as \( \binom{n+k-1}{k-1} \), where \( n \) is the number of ice-creams to choose (6) and \( k \) is the number of types (5). Hence, the expression becomes \( \binom{6+5-1}{5-1} = \binom{10}{4} = 210 \), implying Statement 1 is true.

Step 3: Analyzing Statement 2

Statement 2 relates the problem to arranging 6 A's and 4 B's, treating the selection and non-selection of ice-creams. Here, if we think of arranging 6 A's (chosen) and 4 B's (not chosen) in a row, this describes the number of solutions of the form where A and B represent choices among items. Calculating the arrangement of 10 items (6 A's and 4 B's), we compute \( \binom{10}{4} = 210 \). This shows the statement 2 is also true.

Step 4: Determining if Statement 2 Explains Statement 1

Both statements reflect the same combinatorial concept using different interpretations: choosing 4 positions among 10, which itself is an extrapolation of choosing 6 from 10 as well. Therefore, Statement 2 helps explain why Statement 1 is true as it offers a different viewpoint of the same problem.

Key Concepts

Combination

In combinatorics, a combination is a selection of items from a larger pool, where the order doesn't matter. This is different from permutations, where the order of selection does matter. Combinations focus solely on the selection itself, allowing us to calculate how many different groups can be formed from a set number of items.

To find the number of ways to choose a subset of items from a larger set, we use the combination formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

where \( n \) is the total number of items, \( r \) is the number of items to choose, and \( ! \) denotes factorial, which is the product of all positive integers up to that number.

In our exercise, Statement 1 uses this concept to determine how many ways the child can choose six ice-creams from five types. Using the stars and bars method, we calculated that this is equivalent to choosing 4 positions (or bars) out of 10 (combination of A's and B's), which confirms \( \binom{10}{4} = 210 \).

Stars and Bars Method

The stars and bars method is a popular technique in combinatorics for solving distribution problems. It simplifies problems where you are distributing identical items into different categories or bins.

This method introduces the idea of using stars to represent items and bars to represent dividers between different categories. If you have \( n \) stars to distribute into \( k \) categories, the formula is given by:

• \( \binom{n+k-1}{k-1} \)

This is because you need \((k-1)\) bars to separate \( n + (k-1) \) spaces.

In our example, six ice-creams are represented by stars and the five types of ice-cream by categories. Using stars and bars helps us figure out how to distribute these 6 stars among 5 categories, leading us to the calculation \( \binom{10}{4} \). Both statements in the exercise rely on this technique to reach their conclusions.

Arrangement Problem

Arrangement problems ask us to determine how to order or place items, considering various constraints and scenarios. These problems are key in both mathematics and computer science because they form the basis of understanding permutations and combinations.

Statement 2 in the exercise effectively presents an arrangement problem by comparing the selection of ice-creams to arranging 6 A's and 4 B's in a row. Here, the A's represent chosen ice-creams, and the B's represent the leftover choices.

The focus is on finding different sequences that combine A's and B's, without settling on a fixed starting point for either letter. By arranging 10 positions (6 A's and 4 B's), we compute \( \binom{10}{4} \) for the arrangements and find that there are 210 different ways to do this. This showcases the connection between combination calculations and arrangement solutions in such an exercise.