Problem 97 - A Complete Resource Book in Mathematics for JEE Main

Question

There are four oranges, five apples and six mangoes in a fruit basket. The number of ways in which a person can make a selection of fruits among the fruits in the basket, is (A) 210 (B) 330 (C) 209 (D) None of these

Step-by-Step Solution

Answer

The number of ways to make a selection is 209, which matches option (C).

Step 1: Understanding the Problem

We need to determine how many different ways fruits can be selected from a basket containing four oranges, five apples, and six mangoes. Each fruit can either be selected or not, which means we use combinations to find the total number of selections.

Step 2: Selecting Oranges

From four available oranges, a person can select 0, 1, 2, 3, or 4 oranges. Thus, there are 5 choices (including selecting no oranges at all).

Step 3: Selecting Apples

From five available apples, a person can select 0, 1, 2, 3, 4, or 5 apples. Thus, there are 6 choices.

Step 4: Selecting Mangoes

From six available mangoes, a person can select 0, 1, 2, 3, 4, 5, or 6 mangoes. Thus, there are 7 choices.

Step 5: Calculating Total Selections

The total number of selections is calculated by multiplying the number of options for oranges, apples, and mangoes. Therefore, the total selections are given by: \[5 \times 6 \times 7 = 210.\]

Step 6: Subtracting the Empty Selection

The calculation includes one selection where none of the fruits are selected. To find the number of non-empty selections, subtract this one case: \[210 - 1 = 209.\]

Step 7: Choosing the Final Answer

Since the solution aligns with option (C), the number of ways to make a selection (excluding selecting none of the fruits) is 209.

Key Concepts

Combinations

Combinations are a fundamental concept in combinatorics. They represent the number of ways to choose items from a larger set, where the order does not matter. This is what differentiates combinations from permutations, where order is important.

When dealing with combinations, especially in situations like selecting fruits from a basket, you simply count the different groupings possible without caring about the sequence of selection. For example, selecting 2 oranges from 4 is considered identical whether you pick the first two or the last two.

Mathematically, combinations are often calculated using the binomial coefficient, denoted as \( \binom{n}{r} \). This notation represents the number of combinations of \( n \) items taken \( r \) at a time. It is defined as:

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

This formula ensures that we count the groupings correctly while ignoring any specific order.

Counting Principles

Counting principles allow us to systematically find the number of possible ways to organize or choose objects. The most common counting principle used here is the multiplication principle.

The multiplication principle states that if one event can occur in \( m \) ways and a second event can occur independently of the first in \( n \) ways, then the two events together can occur in \( m \times n \) ways. This principle helps simplify complex counting tasks by breaking them down into simple, manageable parts.

In the context of our fruit basket problem:

• There are 5 ways to choose from the oranges.
• There are 6 ways to choose from the apples.
• There are 7 ways to choose from the mangoes.

By applying the multiplication principle, we compute the total number of ways to select the fruits as \( 5 \times 6 \times 7 = 210 \). This product includes all possible selections, including the empty set where no fruit is chosen.

To find non-empty selections, subtract 1 (the empty set), yielding 209 valid selections. Counting principles streamline such calculations, making them more intuitive.

Permutations

Though permutations might not apply directly to the fruit selection problem since order does not matter here, understanding permutations is crucial in combinatorics. Unlike combinations, permutations consider the arrangement or sequence of selected items.

Permutations are used when the order of selection is important. For example, if you need to arrange 3 letters out of 5 distinct letters, the order you arrange them matters.

Mathematically, permutations of \( n \) items taken \( r \) at a time are denoted as \( P(n, r) \) and calculated by:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

This formula results from counting all possible orders by multiplying continuously decreasing numbers of available items.

Permutations contrast combinations by listing every possible ordered arrangement, which explains why permutations have higher counts compared to combinations for the same number of items and choices. Understanding when to use permutations instead of combinations depends on whether the order of items contributes to different outcomes.