The ice cream shop has 10 types of toppings available, and you decide to add 5 toppings to your bowl of 3 scoops of ice cream. How many combinations of 3 scoops of ice cream and 5 toppings are possible?

To find the total number of combinations of 3 scoops of ice cream and 5 toppings, we can use the fundamental principle of counting, also known as the multiplication principle.

First, we find the number of combinations for the scoops of ice cream:

Since there are 3 scoops of ice cream, and each scoop can be any of the 10 available types, the number of combinations for the scoops of ice cream is [tex]$10^3$.[/tex]

Next, we find the number of combinations for the toppings:

Since there are 5 toppings to choose from, and you're adding all 5 to your bowl, the number of combinations for the toppings is [tex]${10 \choose 5}$,[/tex] which represents choosing 5 toppings out of the 10 available.

Now, to find the total number of combinations, we multiply the number of combinations for the scoops of ice cream by the number of combinations for the toppings:

[tex]\[ \text{Total combinations} = 10^3 \times {10 \choose 5} \][/tex]

Let's calculate this.

We have:

[tex]\[ 10^3 = 10 \times 10 \times 10 = 1000 \]\[ {10 \choose 5} = \frac{10!}{5!(10-5)!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252[/tex]

Now, let's find the total combinations:

[tex]\[ \text{Total combinations} = 1000 \times 252 = 252,000[/tex]

So, there are 252,000 possible combinations of 3 scoops of ice cream and 5 toppings.

Question :

The ice cream shop has 10 types of toppings available, and you decide to add 5 toppings to your bowl of 3 scoops of ice cream. How many combinations of 3 scoops of ice cream and 5 toppings are possible?