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. How many arrangements are possible? What is the probability that each card is next to a card bearing the same numeral?

10 to the tenth power. 10 x 10= 100

10%

I don't understand your work. What is the probability that each card is next to a card bearing the same numeral?

To find the number of possible arrangements, we can consider the total number of arrangements of the ten cards on the table.

Since all ten cards are different, the total number of arrangements is given by the factorial of 10, which is denoted as 10!, and equals:

10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800.

Now let's consider the probability that each card is next to a card bearing the same numeral. To calculate this, we need to determine the favorable outcomes and the total number of outcomes.

Favorable outcomes:
For each numeral from 2 to 6, there are four cards with the same numeral (two spades and two diamonds). These cards can be placed in five possible positions next to each other on the table. Therefore, there are 5 favorable outcomes for each numeral, resulting in a total of 5 x 5 = 25 favorable outcomes.

Total outcomes:
We've already calculated that the total number of arrangements is 10!, which is 3,628,800.

Probability:
The probability is given by the ratio of favorable outcomes to total outcomes.

Probability = (Favorable outcomes) / (Total outcomes)
= 25 / 3,628,800

Simplifying, we find:

Probability ≈ 0.00000689 (rounded to 8 decimal places)

Therefore, the probability that each card is next to a card bearing the same numeral is approximately 0.00000689.