Question

when 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as n 6 7 , where n is a positive integer. what is n?

Answer 1

The probability of getting a sum of 10 when throwing 7 fair standard 6-sided dice is 27/6⁷.

Step-by-step explanation:

The probability of getting a sum of 10 when throwing 7 fair standard 6-sided dice can be written as n 6 7, where n is a positive integer. To find the value of n, we need to determine the number of favorable outcomes, which means finding the number of ways we can get a sum of 10.

One approach is to use the concept of generating functions. We can represent each dice roll as a polynomial with exponents corresponding to the possible outcomes (1, 2, 3, 4, 5, 6). By multiplying these polynomials together, we can find the coefficient of the term with an exponent of 10, which represents the number of ways to get a sum of 10.

Using this method, we find that the coefficient of the term with an exponent of 10 is 27. Therefore, the probability of getting a sum of 10 is 27/6⁷, and n = 27.

Answer 2

To find the probability of rolling a sum of 10 when 7 fair standard 6-sided dice are thrown, we consider the possible combinations and calculate the probability.

Step-by-step explanation:

To find the probability of rolling a sum of 10 when 7 fair standard 6-sided dice are thrown, we need to determine the number of favorable outcomes and the total number of possible outcomes.

Let's consider the possible combinations of numbers on the dice that could sum to 10:

1. (4, 1, 1, 1, 1, 1, 1)

2. (3, 2, 1, 1, 1, 1, 1)

3. (2, 2, 2, 1, 1, 1, 1)

4. (2, 2, 1, 1, 1, 1, 1, 1)

5. (2, 1, 1, 1, 1, 1, 1, 1, 1)

6. (1, 1, 1, 1, 1, 1, 1, 1, 1, 1)

Therefore, there are 6 favorable outcomes.

The total number of possible outcomes can be calculated by multiplying the number of possible outcomes for each dice, which is 6, since each die has 6 sides.

Therefore, the probability can be written as 6/6^7 = 1/6^6 = 1/46656