A person rolls a standard six-sided die 7 times. In how many ways can he get 2 fives, 4 sixes, and 1 two?
The answer is 0 ways. It is impossible to roll 2 fives, 4 sixes, and 1 two with a standard six-sided die.
To find the number of ways a person can get 2 fives, 4 sixes, and 1 two when rolling a six-sided die 7 times, we can use the concept of combinations.
First, let's break the problem down into three parts:
1. The number of ways to get 2 fives: Since there are 6 possible outcomes on each roll and we want to get a five twice, we use the combination formula. The number of ways to choose 2 positions out of 7 for the fives is given by C(7, 2). This can be calculated as 7! / (2! * (7-2)!) = 21.
2. The number of ways to get 4 sixes: Similar to the previous step, we want to choose 4 positions out of 7 for the sixes. Therefore, the number of ways to get 4 sixes is C(7, 4) = 7! / (4! * (7-4)!) = 35.
3. The number of ways to get 1 two: Since we only want one two, we don't need to calculate a combination. There is only one position for the two, which can be placed in any of the 7 rolls.
Now, to find the total number of ways, we multiply the results of each step:
Total number of ways = Number of ways to get 2 fives × Number of ways to get 4 sixes × Number of ways to get 1 two
Total number of ways = 21 × 35 × 7
Total number of ways = 5,985
Therefore, there are 5,985 ways to get 2 fives, 4 sixes, and 1 two when rolling a standard six-sided die 7 times.
To find the number of ways a person can roll 2 fives, 4 sixes, and 1 two on a standard six-sided die 7 times, we can use the concept of permutations.
First, let's calculate the total number of ways to roll a six-sided die 7 times. Since each roll has 6 possible outcomes, the total number of ways is 6^7 (6 raised to the power of 7).
Now we need to determine the number of ways to arrange 2 fives, 4 sixes, and 1 two within those 7 rolls.
Step 1: Choose the positions for the fives
We have 7 rolls, and we need to choose 2 positions for the fives. This can be done using the combination formula: C(n, r) = n! / (r! * (n-r)!), where n is the total number of rolls and r is the number of fives we need to place.
C(7, 2) = 7! / (2! * (7-2)!) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21
Step 2: Choose the positions for the sixes
We have 5 remaining rolls, and we need to choose 4 positions for the sixes.
C(5, 4) = 5! / (4! * (5-4)!) = 5! / (4! * 1!) = (5 * 4 * 3 * 2) / (4 * 3 * 2 * 1) = 5
Step 3: Choose the position for the two
We have 1 remaining roll, and we need to choose 1 position for the two.
C(1, 1) = 1
Step 4: Multiply the number of choices from each step
To find the total number of ways to arrange 2 fives, 4 sixes, and 1 two within 7 rolls, we multiply the choices from each step:
21 * 5 * 1 = 105
Therefore, there are 105 ways to roll a standard six-sided die 7 times and get 2 fives, 4 sixes, and 1 two.