Three fair dice are thrown once. Given that no two dice show the same face.(a) What is the probability that the sum of faces is 7

Sure, I can help with that. Here's how you can find the probability:

(a) The sum of faces is 7

Step 1: Determine the total number of outcomes

When three dice are thrown, each die has 6 possible outcomes. Therefore, the total number of outcomes is 666 = 216.

However, since no two dice can show the same face, we need to adjust this total. For the first die, there are still 6 possible outcomes. For the second die, there are now only 5 possible outcomes (since it can't match the first die). For the third die, there are only 4 possible outcomes (since it can't match the first or second die). Therefore, the total number of outcomes is 654 = 120.

Step 2: Determine the number of outcomes where the sum of faces is 7

The possible combinations where the sum of faces is 7 are: (1,2,4), (1,3,3), (2,2,3), and their permutations.

However, since no two dice can show the same face, we can only consider the combination (1,2,4) and its permutations. There are 3! = 321 = 6 permutations of this combination.

Step 3: Calculate the probability

The probability that the sum of faces is 7 is the number of outcomes where the sum is 7 divided by the total number of outcomes. Therefore, the probability is 6/120 = 0.05.