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ACT Exam  >  ACT Questions  >  Four dice are rolled. The number of possible ... Start Learning for Free
Four dice are rolled. The number of possible outcomes in which at least one dice show 2 is 
  • a)
    1296
  • b)
    671
  • c)
    625
  • d)
    None of these
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Four dice are rolled. The number of possible outcomes in which at leas...
Concept:
Permutations with Repetition = nr
Where n is the number of things to choose from r different things when repetition is allowed, and order matters.
Favorable cases = Total cases - Unfavorable cases
Calculation:
According to the question
Four dies are rolled 
So, Total Possible number of outcomes = 64
Now, Total outcomes when no 2 appears = 54
Now, From the concept used
Favorable cases = 64 - 54
⇒ 1296 - 625
⇒  671
∴ The number of possible outcomes in which at least one die shows 2 is 671.
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Most Upvoted Answer
Four dice are rolled. The number of possible outcomes in which at leas...
Concept:
Permutations with Repetition = nr
Where n is the number of things to choose from r different things when repetition is allowed, and order matters.
Favorable cases = Total cases - Unfavorable cases
Calculation:
According to the question
Four dies are rolled 
So, Total Possible number of outcomes = 64
Now, Total outcomes when no 2 appears = 54
Now, From the concept used
Favorable cases = 64 - 54
⇒ 1296 - 625
⇒  671
∴ The number of possible outcomes in which at least one die shows 2 is 671.
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Community Answer
Four dice are rolled. The number of possible outcomes in which at leas...
To find the number of possible outcomes in which at least one die shows a 2, we can use the principle of complementary counting.

Principle of Complementary Counting:
Instead of directly counting the desired outcomes, we can count the total number of outcomes and subtract the number of outcomes that do not meet the desired condition.

Total Number of Outcomes:
When four dice are rolled, each die has 6 possible outcomes (numbers 1-6). Therefore, the total number of outcomes is 6^4 = 1296.

Number of Outcomes with No 2:
To find the number of outcomes in which no die shows a 2, we can consider each die independently. For each die, there are 5 possible outcomes (numbers 1, 3, 4, 5, and 6) that are not 2. Since there are 4 dice, the total number of outcomes with no 2 is 5^4 = 625.

Number of Outcomes with at Least One 2:
Now, we can use the principle of complementary counting to find the number of outcomes with at least one 2.
Number of outcomes with at least one 2 = Total number of outcomes - Number of outcomes with no 2
= 1296 - 625
= 671

Therefore, the correct answer is option B) 671.
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Directions:In the passages that follow, some words and phrases are underlined and numbered. In the answer column, you will find alternatives for the words and phrases that are underlined. If you think that the original version is best, choose “NO CHANGE,”. You will also find questions about a particular section of the passage, or about the entire passage. These questions will be identified either by an underlined portion or by a number in a box. Look for the answer that clearly expresses the idea, is consistent with the style and tone of the passage, and makes the correct use of standard written English. Read the passage through once before answering the questions. For some questions, you should read beyond the indicated portion before you answer.PassagePuzzling NumbersThe Japanese number game Sudoku has taken the world (1) with storm. While crossword puzzles seem to be the game of choice, (2) they need only sit in an airport or doctor’s office in any city to see how Sudoku is slowly overtaking the popular word game.In the most common game of Sudoku, the player is faced with a nine-by-nine space grid with a total of 81 spaces. The solution to the puzzle requires a number, one through nine, to be placed in each box. (3) Each number must appear only once in each row and only once in each column. A Sudoku puzzle will begin with a few digits already in place. The solver’s (4) task, is to figure out which numeral belongs in the remaining empty spaces. While being a math whiz is no requirement for this seemingly simple process, a level of logic and reasoning (5) is necessary in order to choose the correct number for each space. As with most such activities, practice makes perfect with Sudoku. The puzzles that seem impossible at first glance (6) eventual become more manageable.The name Sudoku is derived from the phrase single number in the Japanese language. (7) Nevertheless, its definition refers to the placement or allotment of a single number. This puzzle is aptly (8) named, since the misplacement of a single number spells the beginning of an incorrect solution. For this reason, Sudoku (9) should have been played using a pencil with a good eraser.It is nearly impossible to discern (10) which number the solver has placed incorrectly once an error is discovered. If you don’t have a pencil, use a pen to write the digit as a series of light dots. This way, (11) should you make a mistake, you can retrace your steps to locate and correct the original errant placement.There are certain strategies involved in solving a Sudoku puzzle, including the process of elimination. For example, if a row of nine spaces already has the number 4, that row cannot contain another 4. This goes for columns as well. The solver can use these clues and a bit of deduction to limit the possible combinations of numbers. By attacking the puzzle using (12) logic, you can turn the process into an amusing challenge rather than an impossible task.1. The origin of Sudoku is attributed to Leonhard Euler, the Swiss mathematician who (13) sometimes developed the game Magic Squares in 1783.2. In fact, Euler’s game was actually based on a game derived from Chinese folklore.3. The primary difference between Magic Squares and Sudoku is that Euler’s game has no grid dividing the puzzle and it is somewhat easier to solve, since there are multiple possibilities for a solution.4.In true Sudoku, each puzzle has only one solution.5. The puzzle’s difficulty varies according to the number of digits that are initially provided.6. When faced with a puzzle with only one 3 provided, for example, you can be sure that it will be more difficult to solve than a puzzle that already has half of the numbers placed in the correct boxes. (14)One has only to examine a local newspaper to ascertain the popularity of this addictive brain teaser. Where comics and crossword puzzles reign supreme, it is often easy to locate a grid of spaces and numbers tucked off in a corner, or even sitting boldly next to the time-honored crossword.Q. (8)Which of the following alternatives to the underlined portion would be LEAST acceptable?

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Directions:In the passages that follow, some words and phrases are underlined and numbered. In the answer column, you will find alternatives for the words and phrases that are underlined. If you think that the original version is best, choose “NO CHANGE,”. You will also find questions about a particular section of the passage, or about the entire passage. These questions will be identified either by an underlined portion or by a number in a box. Look for the answer that clearly expresses the idea, is consistent with the style and tone of the passage, and makes the correct use of standard written English. Read the passage through once before answering the questions. For some questions, you should read beyond the indicated portion before you answer.PassagePuzzling NumbersThe Japanese number game Sudoku has taken the world (1) with storm. While crossword puzzles seem to be the game of choice, (2) they need only sit in an airport or doctor’s office in any city to see how Sudoku is slowly overtaking the popular word game.In the most common game of Sudoku, the player is faced with a nine-by-nine space grid with a total of 81 spaces. The solution to the puzzle requires a number, one through nine, to be placed in each box. (3) Each number must appear only once in each row and only once in each column. A Sudoku puzzle will begin with a few digits already in place. The solver’s (4) task, is to figure out which numeral belongs in the remaining empty spaces. While being a math whiz is no requirement for this seemingly simple process, a level of logic and reasoning (5) is necessary in order to choose the correct number for each space. As with most such activities, practice makes perfect with Sudoku. The puzzles that seem impossible at first glance (6) eventual become more manageable.The name Sudoku is derived from the phrase single number in the Japanese language. (7) Nevertheless, its definition refers to the placement or allotment of a single number. This puzzle is aptly (8) named, since the misplacement of a single number spells the beginning of an incorrect solution. For this reason, Sudoku (9) should have been played using a pencil with a good eraser.It is nearly impossible to discern (10) which number the solver has placed incorrectly once an error is discovered. If you don’t have a pencil, use a pen to write the digit as a series of light dots. This way, (11) should you make a mistake, you can retrace your steps to locate and correct the original errant placement.There are certain strategies involved in solving a Sudoku puzzle, including the process of elimination. For example, if a row of nine spaces already has the number 4, that row cannot contain another 4. This goes for columns as well. The solver can use these clues and a bit of deduction to limit the possible combinations of numbers. By attacking the puzzle using (12) logic, you can turn the process into an amusing challenge rather than an impossible task.1. The origin of Sudoku is attributed to Leonhard Euler, the Swiss mathematician who (13) sometimes developed the game Magic Squares in 1783.2. In fact, Euler’s game was actually based on a game derived from Chinese folklore.3. The primary difference between Magic Squares and Sudoku is that Euler’s game has no grid dividing the puzzle and it is somewhat easier to solve, since there are multiple possibilities for a solution.4.In true Sudoku, each puzzle has only one solution.5. The puzzle’s difficulty varies according to the number of digits that are initially provided.6. When faced with a puzzle with only one 3 provided, for example, you can be sure that it will be more difficult to solve than a puzzle that already has half of the numbers placed in the correct boxes. (14)One has only to examine a local newspaper to ascertain the popularity of this addictive brain teaser. Where comics and crossword puzzles reign supreme, it is often easy to locate a grid of spaces and numbers tucked off in a corner, or even sitting boldly next to the time-honored crossword.Q. (2)

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Four dice are rolled. The number of possible outcomes in which at least one dice show 2 isa)1296b)671c)625d)None of theseCorrect answer is option 'B'. Can you explain this answer?
Question Description
Four dice are rolled. The number of possible outcomes in which at least one dice show 2 isa)1296b)671c)625d)None of theseCorrect answer is option 'B'. Can you explain this answer? for ACT 2025 is part of ACT preparation. The Question and answers have been prepared according to the ACT exam syllabus. Information about Four dice are rolled. The number of possible outcomes in which at least one dice show 2 isa)1296b)671c)625d)None of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for ACT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Four dice are rolled. The number of possible outcomes in which at least one dice show 2 isa)1296b)671c)625d)None of theseCorrect answer is option 'B'. Can you explain this answer?.
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