Tree diagrams/ counting principles/ scp
question: Two cards are drawn in succesion and with out replacement from a deck of 52 cards. find the following
a. the number of ways in which we can obtain the ace of spades and the king of hearts, in that order.
b. The total number of ways in which 2 cards can be dealt.
c. the total number of ways in which 2 cards can be dealt with replacement; that is, the first card is drawn, recorded, and placed back in the deck, and then the second card is drawn and recorded.
I'm not sure what you are asking.
a. There is only one way to first pick an ace of spades and then a king of hearts.
b, c. Which 2 cards? The same ones?
Repost your questions in clearer terms.
page 555
To solve these questions, we can use both tree diagrams and the counting principle. Let's break down each question step by step:
a. The number of ways to obtain the ace of spades and the king of hearts in that order.
To find this, we need to consider the two cards drawn in sequence. Since we draw without replacement, the number of ways to draw the ace of spades and the king of hearts in order is simply 1. The ace of spades is only one card, and once it is drawn, it cannot be drawn again.
b. The total number of ways to deal 2 cards.
In order to calculate this, we need to consider all possible combinations of 2 cards out of the 52-card deck. We can use the formula for combinations, which is nCr = n! / ((r!)*(n-r)!), where n is the total number of items and r is the number of items chosen. In this case, we have n = 52 (the total number of cards) and r = 2 (the number of cards being dealt).
So, applying the formula, we have 52C2 = 52! / ((2!)*(52-2)!) = 52! / (2!*50!) = (52*51) / (2*1) = 1326. Therefore, there are 1326 ways to deal 2 cards from a deck of 52 cards.
c. The total number of ways to deal 2 cards with replacement.
With replacement means that after each card is drawn, it is placed back into the deck, and then another card is drawn. This means that for each card, we have all 52 options available. So, for the first card, we have 52 choices, and for the second card, we again have 52 choices. Since the choices for the two cards are independent, we can multiply the number of choices for each card.
Therefore, the total number of ways to deal 2 cards with replacement is 52 * 52 = 2704.
In summary:
a. There is only 1 way to obtain the ace of spades and the king of hearts in that order.
b. There are 1326 ways to deal 2 cards from a deck of 52 cards.
c. There are 2704 ways to deal 2 cards with replacement from a deck of 52 cards.