A hand consists of 4 cards from a well-shuffled deck of 52 cards. a. Find the total number of possible 4-card poker hands. b. A black flush is a 4-card hand consisting of all black cards. Find the number of possible black flushes. c. Find the probability of being dealt a black flush. a. There are a total of poker hands b. There are possible black flushes. c. The probability is (Round to six decimal places as needed.)

Answer:A) there are 270.725 total of poker handsB) there are 14.950 possible black flushesC) the probability of being dealt a black flush is 0.0552220Step-by-step explanation:Combinations gives the number of ways a subset of r elements can be chosen out of a set of n elements.  Let's use the "n choose r" formula:[tex]nCr=\frac{(n!)}{(r!(n-r)!)}[/tex]A) the total number of combinations of 4 cards chosen from the deck of 52 cards:n = 52r =  4[tex]nCr= \frac{52!}{(4!(52-4)!)} = \frac{52!}{4!(48!)} =\frac{52*51*50*49*48!}{1*2*3*4(48!)}[/tex]The 48! terms cancel[tex]nCr=\frac{52*51*50*49}{1*2*3*4} = \frac{6.497.400}{24}  =270.725[/tex]B) Number of possible black flushes:There are 26 black cards (spades and clubs)n=26r=4[tex]nCr=\frac{26!}{4!(22!)}= \frac{26*25*24*23*22!}{1*2*3*4(22!)}  =\frac{358.800}{24} \\\\nCr= 14.950[/tex]C) Probability of being dealt a black flushSimply divide the result of B) over A) [tex]P=\frac{14.950}{270.725} = 0.0552220[/tex]