Problem 111 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If four whole numbers taken at random are multiplied together, the chance that the last digit in the product is \(1,3,7\), or 9 is (A) \(\frac{4}{625}\) (B) \(\frac{18}{625}\) (C) \(\frac{16}{625}\) (D) none of these

Step-by-Step Solution

Answer

The answer is (C) \(\frac{16}{625}\).

Step 1: Understand the Problem

We need to find the probability that the product of four randomly chosen whole numbers ends in one of the digits 1, 3, 7, or 9.

Step 2: Analyze Individual Digits

Each whole number can end in any digit from 0 to 9 with equal probability. Therefore, there are 10 possible last digits for each of the four numbers.

Step 3: Identify Favorable Last Digits

The product's last digit depends on the last digit of the numbers. To end in 1, 3, 7, or 9, each number should also end in 1, 3, 7, or 9.

Step 4: List All Multiplication Possibilities

Each of the four numbers has 4 options (1, 3, 7, 9) for its last digit. Thus, the total number of favorable outcomes is \(4^4 = 256\).

Step 5: Calculate Total Possible Cases

Since each number has 10 possibilities for its last digit, there are \(10^4 = 10000\) total possible outcomes when multiplying four numbers.

Step 6: Determine Probability

The probability is the ratio of favorable outcomes to total outcomes: \(\frac{256}{10000}\).

Step 7: Simplify the Probability

Simplify \(\frac{256}{10000}\) to \(\frac{16}{625}\) by dividing both the numerator and denominator by 16.

Step 8: Compare with Options

Option (C) \(\frac{16}{625}\) matches the calculated probability.

Key Concepts

Whole Numbers

Whole numbers are a fundamental concept in mathematics. They include all the counting numbers you are familiar with, starting from zero and going upwards: 0, 1, 2, 3, and so on.
This set of numbers does not include any fractions, decimals, or negative values.

When thinking about problems involving whole numbers, here are some useful things to remember:

• Whole numbers are infinite; they go on forever with no upper limit.
• They are all non-negative, meaning you’ll never find a whole number that’s below zero in value.
• A key characteristic of whole numbers is that each has a definite place on the number line, making them easy to visualize.

In probability exercises, like the one we're discussing, whole numbers are often used as the foundation. This ensures calculations are based on a complete and clear set of data.

Multiplication

Multiplication is one of the basic operations in arithmetic, essential in both simple and complex mathematical problems.
It combines groups of equal sizes; for example, multiplying 3 by 4 is like having three groups of four, totaling 12.

Here are some important features of multiplication that can help you understand it's role in probability theory:

• Multiplication is commutative, meaning the order of numbers doesn’t affect the product: for instance, 2 times 3 is the same as 3 times 2.
• This operation is also associative, which allows for grouping numbers in any manner: (2 times 3) times 4 is the same as 2 times (3 times 4).
• In problems where you're calculating probabilities by multiplying whole numbers, the individual probabilities of sequences matter, just as products of numbers determine particular results, such as the last digit in a number.

In the given problem, multiplication helps calculate the product of the last digits of numbers, which directly affects the probability outcome.

Last Digit of a Product

When multiplying numbers, the last digit of their product is influenced solely by the last digits of the factors.
This concept simplifies many calculations, especially when involved in probability exercises.

Here’s how you can think of this:

• The last digit of a number in mathematics is also known as the units place digit.
• To find the last digit of a product, it's necessary to consider only the last digits of each factor; for example, the product of 37 and 42 has the same last digit as the product of 7 and 2.
• In our probability scenario, analyzing the randomness of the digits involved (1, 3, 7, 9) and calculating their combined outcomes illuminate how multiplication affects the probability of certain last digits.

Understanding this concept helps break down complex multiplication problems into manageable parts, focusing on key aspects needed for solving probability exercises effectively.