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Ice cream store two topping on ice cream cone. 106 ways choose topping how many toppings(including no toppings)

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At her favorite ice cream store, Saga can put up to two toppings on an ice cream cone. If there are 106 different ways she can choose toppings (including the choice of no toppings), how many different toppings are there?

May 12, 2018

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106P2=106!/104!=106*105=11130

Or, is it 106C2=106!/2!*104!=106*105/2*1=5565?

Which one?  May 12, 2018
#2
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We can imagine the problem like this:

There are "x" toppings to choose from.

There are $$\binom{x}{2}$$ ways to choose the toppings.

Therefore $$\binom{x}{2}$$ = 106

We can rewrite this as:

$$\frac{x\cdot(x-1)}{2}=106$$

Then we have to add the choice of "no" toppings.

$$\frac{x\cdot(x-1)}{2}+1=106 \\ x^2-x-210=0 \\ (x+14)(x−15)=0 \\ x_1=-14,\ x_2=15$$

Since there cannot be negative toppings, the answer is 15.

We can double check this.

If there are 15 toppings and you can have no toppings, how many ways can we "top" our ice-cream.

$$\binom{15}{2}+1=106$$, it works!

I hope this helped,

gavin

May 12, 2018