To determine the likelihood that a randomly shuffled deck of 52 cards would come together in perfect order by number and suit, we need to consider the number of possible arrangements of the deck and the number of favorable outcomes.
Step 1: Understanding the problem
- A standard deck has 52 cards.
- Each unique order of the deck can be considered one possible arrangement.
- We need to find the probability that the deck is in perfect order, meaning one specific sequence out of all possible sequences.
Step 2: Calculating the total number of possible arrangements
- The total number of possible arrangements of a deck of 52 cards is given by the number of permutations of 52 distinct items. This is denoted by [tex]\(52!\)[/tex] (52 factorial).
- [tex]\(52!\)[/tex] represents the product of all positive integers from 1 to 52.
Step 3: Determining the number of favorable outcomes
- There is exactly one specific arrangement that represents the perfect order by number and suit.
Step 4: Calculating the probability
- The probability ([tex]\(P\)[/tex]) of the deck being in perfect order is the number of favorable outcomes divided by the total number of possible outcomes.
- Since there is only one favorable outcome (the perfect order), the probability is:
[tex]\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{52!}
\][/tex]
Step 5: Expressing the numerical value
- The numerical value of [tex]\( \frac{1}{52!} \)[/tex] is an extremely small number. Using scientific notation, this value has been calculated to be approximately:
[tex]\[
1.2397999308571486 \times 10^{-68}
\][/tex]
This corresponds to the first choice in the given question. Therefore, the likelihood that a randomly shuffled deck of 52 cards would come together in perfect order by number and suit is:
[tex]\[
\frac{1}{52!}
\][/tex]
So, the correct answer is:
[tex]\(\boxed{\frac{1}{52!}}\)[/tex]
