Create an image that visually represents the concept of probability and choice at a restaurant. Display an ideally designed restaurant with a prominent, colorful five-item menu board on the wall, yet make sure it doesn't contain any text. Also, illustrate four distinct customers who are poring over the menu, each deciding on a different meal option. Their choices should be represented by symbolic items (plate with fresh salad, burger, pizza slice, and spaghetti) they are holding, intending to order. The place must be bustling with fresh energy as it's newly inaugurated.

A newly-opened restaurant has 5 menu items. If the first 4 customers each choose one menu item uniformly at random, the probability that the 4th customer orders a previously unordered item is a/b, where a and b are relatively prime positive integers. What is a+b?

let A B C D E be the items

if 4th customer chose A then the other 3 customers can choose other items
B or C or D or E in 4^3 ways
similarly if 4th customer chose B then the others can chose in 4^3 ways
hence the total number of ways in which 4th customer chose one item and the others choose the remaining item =5* 4^3
total ways of choosing 5 items by 4 customers =5^4
therefore the required probability=(5*4^3)/5^4=64/125 .Hence m+n=189

I need a-b

Why did the fourth customer become so indecisive? Because they realized they could choose from 5 different dishes! 🍔🍕🍣

But, let's get down to business. Since the first three customers each chose one item, there are only 2 items left for the fourth customer to select from.

So, the probability that the 4th customer orders a previously unordered item is 2/5.

Therefore, a = 2 and b = 5.

And, a + b = 2 + 5 = 7.

Voilà!

To solve this problem, we need to first determine the total number of possible outcomes and the number of favorable outcomes.

Step 1: Determine the Total Number of Possible Outcomes
Since each customer can choose one of the 5 menu items, the total number of possible outcomes for the first 4 customers is 5 * 5 * 5 * 5 = 625.

Step 2: Determine the Number of Favorable Outcomes
To calculate the number of favorable outcomes, we need to consider two cases:
Case 1: The first three customers each choose a different menu item from each other.
In this case, the 4th customer can choose any of the 5 menu items as they are all previously unordered. The probability of this case is 5 * 4 * 3 = 60.

Case 2: The first three customers have at least one menu item in common.
In this case, the 4th customer needs to choose one of the remaining 4 menu items that have not been ordered by the first three customers. The probability of this case depends on the number of menu items that the first three customers have in common.

The number of ways the first three customers can have at least one menu item in common can be calculated using the principle of inclusion-exclusion. Let's denote the number of ways they can have no common menu items as A, the number of ways they can have one common menu item as B, the number of ways they can have two common menu items as C, and the number of ways they can have three common menu items as D.

We can calculate A as 5 * 4 * 3 = 60 (each of the first three customers can choose one of the 5 menu items, and the second customer has 4 choices since one menu item has already been chosen by the first customer, and so on).
Similarly, we can calculate B as 5 * 4 = 20 (each of the first two customers can choose one of the 5 menu items, and the second customer has 4 choices since one menu item has already been chosen by the first customer).
C can be calculated as 5 * 1 = 5 (the first customer can choose one of the 5 menu items, and the next two customers have only one choice left since two menu items have already been chosen).
Finally, D is simply 5 (since if the first three customers choose the same menu item, there is only one possibility).

Now, using the principle of inclusion-exclusion, the number of ways the first three customers can have at least one menu item in common is A - B + C - D = 60 - 20 + 5 - 5 = 40.

For each of these cases, the 4th customer still has 4 remaining menu items to choose from. Therefore, the number of favorable outcomes for Case 2 is 4 * 40 = 160.

Step 3: Calculate the Probability and Find a/b
The probability is given by the number of favorable outcomes divided by the total number of possible outcomes:
Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
Probability = (160) / (625)
Probability = 32 / 125

Therefore, a = 32 and b = 125.

Step 4: Calculate a + b
a + b = 32 + 125 = 157

So, the value of a + b is 157.

189

189