Ten cards, the 2 through 6 of spades and the 2 through the 6 of diamonds, are shuffled thoroughly and then taken one by one from the top of the deck and placed on the table.
I know how to solve the first question (I think), but how would I solve the second question? (below)
How many arrangements are possible?
10!=10x9x8x7x6x5xx4x3x2x1
What is the probability that each card is next to a card bearing the same numeral?
Whew. when I first read the problem I thought you were trying to find the probability that ANY two pairs were together. But, as I reread the question, you are looking for the probability that ALL pairs are together; much easier.
Choose the first pair to lay down, there are 5 possible pairs, within that pair there are 2 possibilities Spade/diamond or diamond/spade. So (5*2). For the second pair, you have 4 possibilities, again with 2 orders. And so on. So, the number of ways is (5*2)*(4*2)*(3*2)*(2*2)*(1*2)
To solve the second question, let's break it down step by step:
Step 1: Determine the number of favorable outcomes.
Favorable outcomes are arrangements where each card is next to a card bearing the same numeral.
To calculate the number of favorable outcomes, we can arrange each set of cards with the same numeral together. We have 5 sets (2-2, 3-3, 4-4, 5-5, 6-6). The number of ways to arrange each set is 2! (as in each set, the cards can either be in the order of Spades-Diamonds or Diamonds-Spades). Therefore, the number of favorable outcomes is 2! x 2! x 2! x 2! x 2!.
Step 2: Determine the number of total outcomes.
Total outcomes are all possible arrangements of the 10 cards. Since we have already determined that there are 10! possible arrangements, the number of total outcomes is 10!.
Step 3: Calculate the probability.
The probability is the ratio of favorable outcomes to total outcomes.
Probability = Number of Favorable Outcomes / Number of Total Outcomes
Probability = (2! x 2! x 2! x 2! x 2!) / 10!
Now you can calculate the probability using the values we have determined.
Note: Make sure to simplify the fraction before calculating the quotient.
To solve the second question, you would need to determine the number of favorable outcomes and the total number of possible outcomes, and then calculate the probability.
Step 1: Determine the number of favorable outcomes
In this case, a favorable outcome is when each card is placed next to a card bearing the same numeral. So, let's break it down.
For the first card (2 of spades or diamonds), there are no cards to the left, so the only favorable outcome is if the next card is also a 2 (regardless of suit).
For the second card (either the 2 of spades or diamonds, depending on the first card), there are two possibilities for a favorable outcome: if the previous card was also a 2 or if the next card is a 2.
Similarly, for the third, fourth, fifth, and sixth cards, there are also two possibilities for a favorable outcome: either the previous or next card has the same numeral.
Overall, the number of favorable outcomes would be 2^6 = 64 (since there are two possibilities for each of the 6 cards).
Step 2: Determine the total number of possible outcomes
To determine the total number of possible outcomes, you can use the formula for permutations, which is the same as the number of arrangements you mentioned earlier. So, there would be 10! = 3,628,800 possible arrangements.
Step 3: Calculate the probability
Probability is the ratio of favorable outcomes to total outcomes. In this case, the probability would be:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 64 / 3,628,800
Now, you can simplify the fraction or calculate the decimal value for the probability.