TITLE: Combinatorics: Options that do not involve a specific object
QUESTION [1 upvotes]: The question was posed as: Marissa is doing a Tarot reading in which she must pick 6 cards from a deck of 72. The order of their selection is not important. Marissa does not want to see the Fool card. How many of the possible readings do not feature the Fool?
What I think I should do is find the total amount of possibilities of 6 cards out of 72,
C(72,6)= n!
          r!(n-r)!
  =__72!__
     6!(72-6)!

  =__72!__
       6!66! 

  =__6.12e+103_
    (720)(5.44e+92)

  =6.12e+103
     3.92e+95

  =156,238,908 possible combination of cards

Then, find the amount out of a deck of 71
C(71,6) = 71!
           6!65!
    =__8.50e+101__
     (720)(8.25e+90)

    =__8.50e+101__
       5.94e+93

    =143218999

Then subtract them
156238908-143218999= 13019909
That number is way too low, so that's where I am stuck.

REPLY [2 votes]: You're close. You need to find the number of 6 card hands that contain the fool card:
$$\binom{71}{5}$$
Then subtract that number from the total number of 6 card hands in the deck:
$$\binom{72}{6}-\binom{71}{5}$$
{"set_name": "stack_exchange", "score": 1, "question_id": 425857}