Must shuffle the “people inside”... / Derangement

They are the “
□□□ Rangers,” a local mascot group in a certain provincial city.

Born from the personal initiative of the Mayor of □□□, they have become quite well-integrated among the citizens.

Staff members from a department similar to the “Do-It-Now Division,” which once garnered attention, are working to provide citizen services by having the “five rangers” dispatched to various locations and events to address requests submitted by citizens.

However, recently, due to a somewhat sluggish situation and advice from the “Mayor’s daughter,” who is struggling at a startup company while raising children in Tokyo, they are trying to “freshen things up.”

“Nowadays, it’s strange to say blue means male, red means female, pink means a bit sexy... and so on”
The Mayor took the “socially conscious” daughter’s advice seriously and said, “
First, let’s shuffle the ‘people inside each color’!”

At that time, he added the instruction: “
You must change to a color different from your current one and create a new character by fusing ‘each person’s unique character’ with ‘the character each color has formed so far’!”

Then, a staff member knowledgeable in mathematics said, “
Even if you just re-select by lottery, you will have a different color with a probability of roughly ( a )%, so theoretically, if you have them draw lots ( b ) times, you can basically decide on the ‘color change’.”

【Problem】
How many ways are there for “all 5 people to end up with a ‘color’ different from before”? Based on that, find the ‘probability of drawing a different color by lottery,’ round it to an integer, and answer the values for (a) and (b).

※ Regarding the style of the lottery, assume a situation that is “equally likely,” such as “putting 5 colored balls in a bag and having 5 people draw them without replacement.”

【Explanation】
You may have heard of this problem as the “
present exchange problem,” but it is a problem related to what is called a “
derangement
.”

If you are seeing this for the first time, the way to approach it is to “
try thinking about it with a smaller number first,” which will allow you to get an idea of what lies ahead.

① In the case of 2 people, they just need to swap colors with each other, so “
1 way”

② In the case of 3 people, if you think about this in order as well, it is “
2 ways”

③ In the case of 4 people, you can understand this by thinking in order, but let’s organize it. If 4 people A, B, C, and D were originally “A is color a, B is color b, C is color c, D is color d,” and we represent that as (a,b,c,d), the ways to have different colors are (b,a,d,c), (b,d,a,c), (b,c,d,a), (c,d,a,b), (c,a,d,b), (c,d,b,a), (d,c,b,a), (d,c,a,b), (d,a,b,c), and you can see there are “

9 ways

.” (※ At this stage, you should be able to notice the

regularity

!)

④ In the case of 5 people, you can count them by thinking in order as shown above, but if you can notice the regularity, you can find it by calculation alone: ∴ “4 × (2 + 9) =

44 ways.” (※ If you don’t understand, please look up ‘derangement’.)

Here, the total number of cases is "5!=5×4×3×2×1=120 ways", so the
probability is "
44/120≒0.366..."(*). This results in.

When this number of people becomes ∞, the probability becomes "
1/e≒0.367... (← something to look forward to in high school curriculum!)", but even from the result of (*), we can derive "∴
a=37,b=3
", can't we?