Answer:
Therefore, the number of 4-card hands that can be formed to have two pairs of cards is 261,456.
Explanation:
To calculate the number of 4-card hands that can be formed with two pairs of cards from a deck of 52 cards, we can break down the problem into steps:
Step 1: Select the two ranks for the pairs.
There are 13 ranks in a standard deck of cards, so we can choose 2 ranks out of the 13 available. This can be calculated using combinations: C(13, 2) = 13! / (2! * (13-2)!) = 78.
Step 2: Select two cards of the first rank.
For each pair rank, we have four cards of that rank in the deck. So we need to choose 2 cards out of the 4 available for the first rank: C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 3: Select two cards of the second rank.
Similar to the previous step, we need to choose 2 cards out of the 4 available for the second rank: C(4, 2) = 6.
Step 4: Select the remaining two cards from the remaining 44 cards.
After selecting the two pairs, we are left with 48 cards in the deck (52 - 4 - 4 = 44). We need to choose 2 cards from the remaining 44: C(44, 2) = 44! / (2! * (44-2)!) = 946.
Step 5: Multiply the results of all steps.
To find the total number of 4-card hands with two pairs, we multiply the results of all steps together:
78 * 6 * 6 * 946 = 261,456.