Question

Four dice (Six-faced) are rolled. The number of possible outcomes in which at least one die shows $2$ is
1. $1296$
2. $625$
3. $671$
4. None of these

Answer 1

Hint: In this question, we are given that four dice are rolled and we have to find the number of outcomes that at least one die shows $2$. The first step is to find the total number of outcomes using the formula ${6^n}$ where $n = 4$. Now find the outcomes if none of the dice will show $2$ and subtract them to calculate the required outcomes.

Formula Used:
Number of outcomes when a die is rolled $ = {6^n}$
Here, $n$ is the number of dice rolled

Complete step by step Solution:
Given that, four dice are rolled
Total number of possible outcomes when four dice are rolled $ = {6^4} = 1296$
Let the condition be none of the dice will show $2$,
Then the outcomes will be ${5^4} = 625$
Therefore, the number of possible outcomes in which at least one die shows $2$ will be the difference between total outcomes and the outcomes that none of the dice will show $2$
$ = 1296 - 625$
$ = 671$
Hence, Option (3) is the correct answer i.e., $671$.

Hence, the correct option is 3.

Note: The key concept involved in solving this problem is a good knowledge of probability and outcomes. Students must know that to get the total number of outcomes, simply multiply the events together. For example, flipping a coin has two possible outcomes, whereas rolling a die has six possible outcomes. We can get the total number of outcomes for the sample space by multiplying them together.