A poker hand is defined as drawing 5 cards at random without replacementfrom a deck of 52 playing cards. Find the probability of each of the following poker hands:(a) Four of a kind (four cards of equal face value and one card of a different value).(b) Full house (one pair and one triple of cards with equal face value).(c) Three of a kind (three equal face values plus two cards of different values).(d) Two pairs (two pairs of equal face value plus one card of a different value).(e) One pair (one pair of equal face value plus three cards of different values).

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Sure, let's calculate the probabilities for each of these poker hands:

(a) Four of a kind: This means you have 4 cards of the same rank and one card of a different rank. There are 13 different ranks and for each rank, there are 4 cards. So, there are 13 ways to choose the rank of the 4 cards and 48 ways to choose the other card. The total number of ways to choose 5 cards from 52 is 52 choose 5. So, the probability is (13 * 48) / (52 choose 5).

(b) Full house: This means you have 3 cards of one rank and 2 cards of another rank. There are 13 ways to choose the rank of the 3 cards, 4 choose 3 ways to choose the 3 cards of that rank, 12 ways to choose the rank of the other 2 cards, and 4 choose 2 ways to choose the 2 cards of that rank. So, the probability is (13 * (4 choose 3) * 12 * (4 choose 2)) / (52 choose 5).

(c) Three of a kind: This means you have 3 cards of one rank and 2 cards of different ranks. There are 13 ways to choose the rank of the 3 cards, 4 choose 3 ways to choose the 3 cards of that rank, 12 choose 2 ways to choose the ranks of the other 2 cards, and 4 ways to choose each of those cards. So, the probability is (13 * (4 choose 3) * (12 choose 2) * 4 * 4) / (52 choose 5).

(d) Two pairs: This means you have 2 cards of one rank, 2 cards of another rank, and 1 card of a different rank. There are 13 choose 2 ways to choose the ranks of the pairs, (4 choose 2)^2 ways to choose the 2 cards of each rank, 11 ways to choose the rank of the other card, and 4 ways to choose that card. So, the probability is ((13 choose 2) * (4 choose 2)^2 * 11 * 4) / (52 choose 5).

(e) One pair: This means you have 2 cards of one rank and 3 cards of different ranks. There are 13 ways to choose the rank of the pair, 4 choose 2 ways to choose the 2 cards of that rank, 12 choose 3 ways to choose the ranks of the other 3 cards, and 4^3 ways to choose each of those cards. So, the probability is (13 * (4 choose 2) * (12 choose 3) * 4^3) / (52 choose 5).