Question

May i please get help with this? Thank you for your time.

[Poker] Poker is a card played with a standard 52 card playing deck (https://en.wikipedia.org/wiki/Standard_52-card_deck). There are many variants of poker. However, for this problem we will not concern ourselves with any specific version, but rather the types of poker hands possible. Assume five cards out of the standard 52 card deck are dealt to a player. The order that the player receives the cards does not matter. One of the hand types is a three-of-a-kind. In this hand a player has 3 of the same rank of card, a card of a second rank, and a card of a third rank. The two single cards are commonly referred to as "kickers". For instance, the following are all three of a kind (Note: Qh in the following example denotes Queen of Hearts, 3s denotes 3 of Spades, etc): Qh Qs Qc 7c 5d (the 7 of clubs and 5 of diamonds are the kickers) 5d 5c 5s 3s 2c 7s 7c 7d Ks As The following is NOT a three of a kind: 7s 7d 7c Qs Qd (There's a group of 3 and 2 here. Must be a group of 3, 1, 1) How many unique ways can a poker player make a three-of-a-kind if both kicker cards are clubs?

Answer 1

Solution :- The answer for the given question is 3432.

Following is the explanation about the same.

$\small \\ 1. \textrm{Ways\ of\ selecting\ the\ rank\ of\ three-of-a-kind is :} \binom{13}{1}$

$\small \\ 2. \textrm{Ways of selecting suits of the three-of-a-kind is :} \binom{4}{3}$

$\small \\ 3. \textrm{Ways of selecting two ranks (kicker cards) from the remaining twelve ranks (from clubs only) is :} \binom{12}{2}$

(Note :- Step 3 shows we are selecting 2 ranks from remaining 12 ranks. But suit is fixed as clubs for both kicker cards.)

$\small \\ \textrm{Unique ways are} \binom{13}{1} * \binom{4}{3} * \binom{12}{2} = 13 * 4 * 66 = 3432$

Hence, we can observe that there are 3432 unique ways for the given conditions.