Consider 8 card hands dealt from a standard deck (1)How many 8 card hands have 5

Question

Consider 8 card hands dealt from a standard deck

(1)How many 8 card hands have 5 cards of one suit and 3 of another.
(2) How many 8-card hands have 4 cards of one suit and 4 of another.
(3) How many 8 card hands have all 8 cards with numbers in (11,12,13) where 11 , 12, 13 are Jack Queen King.
(4) How many 7 cards with J,Q,K and one card with a number 1-10 can you have?
(5) How many have cards with 8 different numbers?
(6) How many have two 3 of a kinds and a pair? (3 of number 3 of number B 2 of number C) where they don't equal each other?
7.) How many 8 card hands have at least one card with the nunber one or Ace

Explanation / Answer

According to rules of chegg, only first four questions are need to be answered unless specified. So, next time please mention in the questions, request to do all.

1) There are 4 suits each 13 cards

Hence, first select 2 suits out of 4

= 4 C 2

Next in each suit, select 5 cards from one suit and 3 cards from another

= 13 C 5 * 13 C 3

Hence total answer = multiply all above = 6 * 1287 * 286 = 2208492

2) 4 cards from one suit and 4 cards from another = 4C2 * 13C4* 13C4 = 3067350

3) first select jack, queen, 11, 12, 13 and then select 8 cards of 20 cards= 20C8 = 125970

4) 7 cards from J,Q,K = 12C7

1 card from 1-10 = 40 C 1

total = 792 * 40 = 31680

5) Cards with 8 different nos = 10 C 8 * 4C1 * 4C1 * 4C1 * 4C1 ...8times (ACE is taken as 1)

= 184320

7) 8 cards selected with atleast one ace

= 4C1* 48C7+ 4C2* 48C6 + 4C3* 48C5 + 4C4* 48C4

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