Permutation Conceptual Understanding Question

[user2833]

Hey guys, I hope you are all having a delightful week.

I was doing the “Multiplication: The Choice Method” video and had a question on the last problem in that video. The questions is as follows: An ice cream shop has 10 different flavors of ice cream and 6 additional toppings. If a customer must get at least one topping, how many ice concoctions can one make?

Greg answers the question as (10 * 2 * 2 * 2 * 2 * 2 * 2) - 1 = 639.

My thinking was that if you have 10 flavors, and 6 toppings, where you have to choose at least one, it would be 10 flavors * 6 choices * yes/no * yes/no * yes/no * yes/no * yes/no * yes/no (so 5 yes/no choices after the first topping choice), for an answer of 1920 (10*6*2*2*2*2*2). This clearly, is wrong. Would you be able to explain why? Intuitively, I feel my answer makes sense. I have attached a screen shot of the question for reference.

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[Leaderboard]

BTW the answer is 630 not 639 - see the note below the video.

Also: the issue with your approach is that it’s not like you have 6 choices for that first topping. The first topping also is a yes/no.

[user2833]

So the way I thought about it was that for each flavor of ice cream, you have 6 options for your first choice, and then the rest are entirely optional, so it would be 5 further yes no questions?

For instance if you have toppings: A, B, C, D, E and F. Then after choosing B, the rest are optional. I feel like I’m just rewording my initial statement, but I’m not sure I understand your response entirely. Could you elaborate?

[Leaderboard]

Suppose you take topping 2, for instance. You would both be saying “yes” to the yes/no case and including this in the “6 choices” part = duplicate.