Probability problems (Permutations, Combinations, law of inclusion and exclusion)

We have only covered basic concepts so the solutions are more about proper combinatorics and permutations. As well as law of inclusion and exclusion. Provide the solutions. Also briefly explain (in layman's terms) the solutions to each problem (sepately).  Please write clearly. "Method at the top" are the 3 steps outlined on top of the first page 

https://imgur.com/a/hsqmZca 

Probability
Ugher Ugher
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Mathe Mathe
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  • Ugher Ugher
    0

    Thank you. This is excellent. Can you explain this trivial answer I came across ? If the card deck (52 cards) is shuffled, what is the chance that the entire suit of hearts is together in the deck ? The answer states (40!)(13!)/(52!) but why is it 40 and not 39 ? Since there are 52 cards and 13 cards in a suit it should be (39!)(13!)/(52!), no ?

    • Mathe Mathe
      0

      The 13 heart cards could be together starting position 1 or 2 or 3 or position 40. In other words, consider all the possible placements in which the hearts can be together. You must multiply by that number (40), making 39! into 40!. A tip would be appreciated!

  • Ugher Ugher
    0

    I will tip. But I still don't quite understand why a sequence of 13 (suit of hearts, let's call it a "chunk") must start at 1 or 2 or 3 or 40. I would understand if you said that they must start at any position that has atleast 13 spots left (52-13) is 39 and not 40. I still don't get it.

    • Mathe Mathe
      0

      If they chunk is together, we have too choose where to place it among the 52 cards. There are 40 places we can put the chunk together.

  • Ugher Ugher
    0

    I dont understand where the 40 comes from

    • Mathe Mathe
      0

      The chunk can go in one of 40 places. Consider 5 cards total and a chunk of size two. Where are you placing the chunk? At they very beginning, in second place or in third place?

  • Ugher Ugher
    0

    I still don't get it. But I tipped you for good work regardless.

    • Ugher Ugher
      0

      can I ask you a direct question ? I need to have my test reviewed and corrected, and I liked the style of your solutions. It's 5 questions and they are pretty short but I need you to type out a clear solution and point out what I did wrong

    • Mathe Mathe
      0

      Yes you can. I can read it and begin reviewing it in an hour. The system allows for direct questions.

  • Ugher Ugher
    0

    alright I will post right now

  • Ugher Ugher
    0

    Thank you, I accepted the anwer, I will ask maybe 2-3 follow up questions later.

The answer is accepted.
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