Pizza Overload
John goes to a pizzeria which offers 15 toppings. Also, for the customers who would like to taste 2 different toppings while buying a single pizza, they have an option of 'half-n-half' where your pizza can have a combination of two different toppings, say pepperoni and Margarita. Half of the pizza will be having a Margherita topping and half will be having Pepperoni. This is a considered to be a separate topping in itself. Margherita , pepperoni and the combo of half margherita and half pepperoni are considered as 3 different toppings. (Note: the 15 toppings available are 'single' toppings. You can choose from those 15 or create a new topping of your choice which would be a combination of 2 toppings on the same pizza, half of pizza having topping A and half having topping B). John buys 6 pizzas. (John can also buy pizza with exact same topping more than once) . Let the total number of possibilities be .
Find the value of .
The answer is 46906255.
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1 solution
Total number of possible Combinations of half n half = 15C2. Thats 105 pizzas with a combination of 2 toppings (half-n-half). There are 15 toppings. Thus 15 pizzas with single toppings. Total 120 possible toppings (15 normal i.e. 'pure' toppings or single toppings and 105 half-n-half toppings). All 120 toppings are different. Required Solution is the number of non negative integral solutions of x1 + x2 + x3+ ... + x120 = 6. Where x1, x2, x3, ... x120 represent the 120 different toppings. Thats equal to 125C6. 125C6 = 4690625500. Divide by 100. This gives 46906255. This already is an integer and for an integer p, [p] = p itself. And we get the final answer as 46906255.