jonathanldong

If order doesn't matter, then there are C(12, 3) choices, which is \((12 \cdot 11 \cdot 10) \div (3 \cdot 2 \cdot 1)=1320 \div 6=220.\)

However, if order does matter, we count the amount of choices for the first option:12. The second option has 11 choices, and the third, 10. we multiply them to get 1320 choices.

I found the answer. the sum of those numbers are 409158, so the digit is 8.

sorry, incorrect

Divide it into triangles and rectangles and add the area of those up

I believe you have left something out, as you did not state the total amount of flavors. For your reference, you can find the answer by finding C(total amount of flavors, 3)

I found this page on it: https://brilliant.org/wiki/cardano-method/. If your intrested you can go check it out. This offers a solution to the full cubic by first depressing it and then solving that.

so we know that these four quarters of circles make up a full circle with a radius of 4 cm. So we need to find 36-(3.14*16-overlaping region). So the actual problem is finding the overlapping region. We know that the straight line under the overlapping part is 2, as 4+4-6=2. Can you carry on from there?

I think you have to find the areas of the right triangles surrounding the grey square and use that to find the ratio. You have to use the Pythagorean theorum to find the length of the slanted line and find the part which is the length of the grey square

I think this is a type of equation called a depressed cubic.

oh yeah I forgot the 1. Thanks, Cphill!

For your second post:

1. n must be a negative number, as all positive values of n would be greater than n-3. n must also be greater than -6, as -6-3=-9, and the absolute value of that would be nine, so 0>n>-6, so the possible integer values of n are 0, -1, -2, -3, -4, and -5. Add that up, and you get -15.