A standard 52-card deck contains four queens, twelve face cards, thirteen hearts (all red), thirteen diamonds (all red), thirteen spades (all black), and thirteen clubs (all black). Of the 752,538,150 different eight-card hands possible, decide how many would consist of the following. (a) all clubs (b) all black cards (c) all queens (a) There are 1287 ways to have a hand with all clubs. (Simplify your answer.) (b) There are 1562275 ways to have a hand with all black cards. (Simplify your answer.) (c) There are $\square$ ways to have a hand with all queens. (Simplify your answer.)

Step 1 ：A standard 52-card deck contains four queens, twelve face cards, thirteen hearts (all red), thirteen diamonds (all red), thirteen spades (all black), and thirteen clubs (all black). Of the 752,538,150 different eight-card hands possible, decide how many would consist of the following.

Step 2 ：(a) There are 1287 ways to have a hand with all clubs.

Step 3 ：(b) There are 1562275 ways to have a hand with all black cards.

Step 4 ：The question asks for the number of ways to have a hand with all queens. There are 4 queens in a deck of 52 cards. The number of ways to choose all 4 queens from 4 is simply 1, because there's only one way to choose all items from a set: choosing all of them. However, we are choosing 8 cards in total, so the remaining 4 cards can be any of the remaining 48 cards in the deck.

Step 5 ：The number of ways to choose 4 cards from 48 is given by the combination formula 'n choose k', which is \(\frac{n!}{k!(n-k)!}\), where n is the total number of items, k is the number of items to choose, and '!' denotes factorial.

Step 6 ：Let n = 48 and k = 4, the number of ways to choose 4 cards from 48 is 194580.

Step 7 ：Final Answer: There are \(\boxed{194580}\) ways to have a hand with all queens.