A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, c

Question

A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called ‘hearts’ and ‘diamonds’) are red, the other two ($\spadesuit$ and $\clubsuit$, called ‘spades’ and ‘clubs’) are black. The cards in the deck are placed in random order (usually by a process called ‘shuffling’). In how many ways can we pick two different cards

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Melanie 3 months 2022-02-03T03:56:44+00:00 1 Answer 0 views 0

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    2022-02-03T03:57:48+00:00

    Answer:

    The number of ways to select 2 cards from 52 cards without replacement is 1326.

    The number of ways to select 2 cards from 52 cards in case the order is important is 2652.

    Step-by-step explanation:

    Combinations is a mathematical procedure to compute the number of ways in which k items can be selected from n different items without replacement and  irrespective of the order.

    [tex]{n\choose k}=\frac{n!}{k!(n-k)!}[/tex]

    Permutation is a mathematical procedure to determine the number of arrangements of k items from n different items respective of the order of arrangement.

    [tex]^{n}P_{k}=\frac{n!}{(n-k)!}[/tex]

    In this case we need to select two different cards from a pack of 52 cards.

    • Two cards are selected without replacement:

    Compute the number of ways to select 2 cards from 52 cards without replacement as follows:

    [tex]{n\choose k}=\frac{n!}{k!(n-k)!}[/tex]

    [tex]{52\choose 2}=\frac{52!}{2!(52-2)!}[/tex]

          [tex]=\frac{52\times 51\times 50!}{2!\times50!}\\=1326[/tex]

    Thus, the number of ways to select 2 cards from 52 cards without replacement is 1326.

    • Two cards are selected and the order matters.

    Compute the number of ways to select 2 cards from 52 cards in case the order is important as follows:

    [tex]^{n}P_{k}=\frac{n!}{(n-k)!}[/tex]

    [tex]^{52}P_{2}=\frac{52!}{(52-2)!}[/tex]

           [tex]=\frac{52\times 51\times 52!}{50!}[/tex]

           [tex]=52\times 51\\=2652[/tex]

    Thus, the number of ways to select 2 cards from 52 cards in case the order is important is 2652.

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