Hello,

I have a probability problem on combination that is not easy to solve for me, and I need help. Thank you.
Probleme:

Problem:
Little Caesar's Pizza had a commercial on tv that said you could buy 1 pizza and get another one free, with up to 5 toppings per pizza. The dialogue went something like this: The older person said, " 5 toppings per pizza , that is 10 different pizzas to choose from"
The little boy then said "no, it is 1,048,576 combinations" Do you think the little boy is right? show the operations to find the number of combinations.

There are two ways to look at this:

Put the two pizza's side by side, and offer the 5 toppings to each, one at a time. The customer has a choice of "yes" or "no" to each topping, ten times, so the number of ways the two pizza's can be made is at most 2^10=1024.

Since the two pizza's can be swapped, so we over-counted by 2, and the number of ways to order the two pizza's drop to 1024/2=512.

The other way is to calculate the number of ways we can order 1 pizza, which equals 2^5=32 by the same reasoning above. There are 32^2 ways to order two pizzas, except for the duplication. So the number of ways to order the two pizza would be 32^2/2=512, again.

thank you for your help.I really appreciate it.

You're welcome!

To determine whether the little boy is right about the number of combinations, we need to calculate the total number of combinations for the given situation.

The problem involves buying one pizza and getting another one free, with a maximum of 5 toppings per pizza.

To calculate the number of combinations, we'll analyze it step by step:

Step 1: Count the number of possible topping choices for each pizza.
Since there are 5 toppings per pizza, we have 5 choices for the first pizza and 5 choices for the second pizza.

Step 2: Calculate the number of combinations for each topping choice.
Let's assume we have T1, T2, T3, T4, and T5 as the five topping choices for each pizza.

For the first pizza, we can have any combination of these toppings, ranging from 0 to 5 toppings. This can be calculated using the formula for combinations: nCr = n! / ((n-r)! * r!).

We need to consider all possible combinations of 0 to 5 toppings, so we need to calculate:
C0 + C1 + C2 + C3 + C4 + C5.

Using the combination formula for each value of r, we can calculate the number of combinations for each topping choice:

C0 = 1
C1 = 5C1 = 5
C2 = 5C2 = 10
C3 = 5C3 = 10
C4 = 5C4 = 5
C5 = 5C5 = 1

Now, we add up all the combinations:
C0 + C1 + C2 + C3 + C4 + C5 = 1 + 5 + 10 + 10 + 5 + 1 = 32.

Therefore, for the first pizza, there are 32 different combinations.

Step 3: Multiply the number of combinations for each pizza.
Since there are two pizzas, we need to multiply the number of combinations for the first pizza (32) by the number of combinations for the second pizza (also 32).

32 * 32 = 1024.

So, the total number of combinations for buying 1 pizza and getting another one free with up to 5 toppings per pizza is 1024, not 1,048,576 as the little boy claimed.

Therefore, the little boy is incorrect in this case.