A card game uses a pack of 24 cards, consisting of ace, king, queen, jack, 10, and 9 in each of the four suits spade, hearts, diamonds and clubs. A hand consist of 6 cards. Find the probability a hand includes all four aces.

There are 24C6 different hands. That's 134,596.

Given 4 aces, there are 20C2 ways to pick the other two cards, That's 190.

So, p = 190/134596 = 0.0014

To find the probability of a hand including all four aces, we need to determine the number of favorable outcomes (hands that include all four aces) and the total number of possible outcomes (all possible hands).

First, let's consider the number of favorable outcomes. There are 4 aces in the deck, and we need all four to be present in the hand. Since the hand consists of 6 cards, we have 6 slots to place the four aces. We can choose any 4 slots out of the 6 to place the aces. This can be calculated using combinations.

The number of combinations of choosing 4 slots out of 6 is denoted as C(6, 4), which is calculated as:

C(6, 4) = 6! / (4!(6-4)!) = 6! / (4!2!) = (6 * 5 * 4!)/(4!2!) = 15

So, the number of favorable outcomes is 15.

Next, let's determine the total number of possible outcomes (all possible hands). Since we have a deck of 24 cards and the hand consists of 6 cards, the total number of possible outcomes can be calculated as the number of combinations of choosing 6 cards out of 24, denoted as C(24, 6).

C(24, 6) = 24! / (6!(24-6)!) = 24! / (6!18!) = (24 * 23 * 22 * 21 * 20 * 19)/(6 * 5 * 4 * 3 * 2 * 1) = 13459560

So, the total number of possible outcomes is 13,459,560.

Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:

Probability = Number of favorable outcomes / Total number of possible outcomes
= 15 / 13,459,560

Therefore, the probability of a hand including all four aces is approximately 0.000001116, or about 0.0001%.