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$The digits 1, 2, 3, 4, 5, and 6 are randomly arranged to form a three-digit number. (Digits are not repeated.) Find the probability that the number is even and greater than 600 . The probability that the three-digit number is even and greater than 600 is (1)/(10). (Type an integer or a simplified fraction.)$

The digits $1$ , $2$ , $3$ , $4$ , $5$ , and $6$ are randomly arranged to form a three-digit number. (Digits are not repeated.) Find the probability that the number is even and greater than $600$ . $\newline$ The probability that the three-digit number is even and greater than $600$ is $\frac{1}{10}$ . $\newline$ (Type an integer or a simplified fraction.)

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Math Problems

Grade 7

Understanding integers

Full solution

Q. The digits

1

2

3

4

5

, and

6

are randomly arranged to form a three-digit number. (Digits are not repeated.) Find the probability that the number is even and greater than

600

\newline

The probability that the three-digit number is even and greater than

600

\frac{1}{10}

\newline

(Type an integer or a simplified fraction.)

Total Possible Numbers: First, let's determine the total number of possible three-digit numbers that can be formed using the digits $1, 2, 3, 4, 5,$ and $6$ without repetition.
Even Numbers Criteria: There are $6$ choices for the first digit, $5$ choices for the second digit, and $4$ choices for the third digit, since we cannot repeat digits.
Even Numbers Greater Than $600$ : The total number of possible three-digit numbers is therefore $6 \times 5 \times 4 = 120$ .
Calculate Probability: Now, let's find the number of three-digit numbers that are even. A number is even if its last digit is even, which in this case can be $2$ , $4$ , or $6$ .
Calculate Probability: Now, let's find the number of three-digit numbers that are even. A number is even if its last digit is even, which in this case can be $2$ , $4$ , or $6$ .For a number to be greater than $600$ , the first digit must be $6$ . So, we have only $1$ choice for the first digit.
Calculate Probability: Now, let's find the number of three-digit numbers that are even. A number is even if its last digit is even, which in this case can be $2$ , $4$ , or $6$ .For a number to be greater than $600$ , the first digit must be $6$ . So, we have only $1$ choice for the first digit.Since the last digit must be even, we have $3$ choices for the last digit ( $2$ , $4$ , or $6$ ), but we've already used the digit $6$ for the first digit, so we actually have only $2$ choices for the last digit ( $2$ or $4$ ).
Calculate Probability: Now, let's find the number of three-digit numbers that are even. A number is even if its last digit is even, which in this case can be $2$ , $4$ , or $6$ .For a number to be greater than $600$ , the first digit must be $6$ . So, we have only $1$ choice for the first digit.Since the last digit must be even, we have $3$ choices for the last digit ( $2$ , $4$ , or $6$ ), but we've already used the digit $6$ for the first digit, so we actually have only $2$ choices for the last digit ( $2$ or $4$ ).For the second digit, we have $4$ remaining choices (since we cannot use the digit chosen for the first digit or the last digit).
Calculate Probability: Now, let's find the number of three-digit numbers that are even. A number is even if its last digit is even, which in this case can be $2$ , $4$ , or $6$ .For a number to be greater than $600$ , the first digit must be $6$ . So, we have only $1$ choice for the first digit.Since the last digit must be even, we have $3$ choices for the last digit ( $2$ , $4$ , or $6$ ), but we've already used the digit $6$ for the first digit, so we actually have only $2$ choices for the last digit ( $2$ or $4$ ).For the second digit, we have $4$ remaining choices (since we cannot use the digit chosen for the first digit or the last digit).The number of even three-digit numbers greater than $600$ is therefore $4$ $6$ .
Calculate Probability: Now, let's find the number of three-digit numbers that are even. A number is even if its last digit is even, which in this case can be $2$ , $4$ , or $6$ .For a number to be greater than $600$ , the first digit must be $6$ . So, we have only $1$ choice for the first digit.Since the last digit must be even, we have $3$ choices for the last digit ( $2$ , $4$ , or $6$ ), but we've already used the digit $6$ for the first digit, so we actually have only $2$ choices for the last digit ( $2$ or $4$ ).For the second digit, we have $4$ remaining choices (since we cannot use the digit chosen for the first digit or the last digit).The number of even three-digit numbers greater than $600$ is therefore $4$ $6$ .To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes.
Calculate Probability: Now, let's find the number of three-digit numbers that are even. A number is even if its last digit is even, which in this case can be $2$ , $4$ , or $6$ .For a number to be greater than $600$ , the first digit must be $6$ . So, we have only $1$ choice for the first digit.Since the last digit must be even, we have $3$ choices for the last digit ( $2$ , $4$ , or $6$ ), but we've already used the digit $6$ for the first digit, so we actually have only $2$ choices for the last digit ( $2$ or $4$ ).For the second digit, we have $4$ remaining choices (since we cannot use the digit chosen for the first digit or the last digit).The number of even three-digit numbers greater than $600$ is therefore $4$ $6$ .To find the probability, we divide the number of favorable outcomes by the total number of possible outcomes.The probability is $4$ $7$ (favorable outcomes) divided by $4$ $8$ (total possible outcomes), which simplifies to $4$ $9$ .