Question

Ice Cream Scoops - In shops with lots of ice-cream flavors there are many different flavor combinations, even with only a 2-scoop cone. With 1 ice-cream flavor there is 1 kind of 2-scoop ice cream, but with 2 flavors there are 3 possible combinations (eg vanilla/vanilla, chocolate/chocolate, and vanilla/chocolate). How many possible 2-scoop cone combinations are there with 3 flavors? How about 4 flavors? 5 flavors? Use your results to develop a pattern or formula and use it to predict the number of 2-scoop combinations with 10 flavors.

Answer 1

Answer:

The general formula that can be used to find the required combinations for n flavor ice-cream.

$N_n = \frac{n(n+1)}{2}$

For 10 ice-cream flavors

$N_{10} = \frac{10(10+1)}{2}$

$N_{10} = 55$

Step-by-step explanation:

Let V = Vanila, C = Chocolate, S = Strawberry, P = Pineapple, B = Berry

2 ice-cream flavor:

V, C

{V/V, V/C, C/C}

3 possible combinations

3 ice-cream flavor:

V, C, S

{V/V, V/C, V/S, C/S, C/C, S/S}

6 possible combinations

4 ice-cream flavor:

V, C, S, P

{V/V, V/C, V/S, V/P, C/S, C/P, S/P, C/C, S/S, P/P}

10 possible combinations

5 ice-cream flavor:

V, C, S, P, B

{V/V, V/C, V/S, V/P, V/B, C/S, C/P, C/B, S/P, S/B, P/B, C/C, S/S, P/P, B/B }

15 possible combinations

So we have a series of

3, 6, 10, 15...

This series is known as Triangular number series

1 + 2 = 3

1 + 2 + 3 = 6

1 + 2 + 3 + 4 = 10

1 + 2 + 3 + 4 + 5 = 15

The general formula for this series is

$N_n = \frac{n(n+1)}{2}$

Using the above formula we can predict the number of 2-scoop combinations with 10 flavors.

$N_{10} = \frac{10(10+1)}{2}$

$N_{10} = 55$

Therefore, there are 55 different combinations of 2-scoop  with 10 flavors.