Question

An ice cream parlor has 6 flavors of ice cream.  A double-scoop cone can have any 2 flavors, including the same flavor twice.  How many different double-scoop combinations are possible? Please explain in detail.

Answer 1

Double-Scoop Combinations

To find out the number of different double-scoop combinations possible with 6 flavors of ice cream, where the combinations can include the same flavor twice, we use the formula for combinations with replacement:

The formula for combinations with replacement is:

`$\[C(n + r - 1, r) = ((n + r - 1)!)/(r! (n - 1)!)\]$`

Where:

• 
  `\( n \)` is the total number of flavors.
• 
  `\( r \)` is the number of scoops to choose.

In this case,
`\( n = 6 \)` and
`\( r = 2 \)`. Plugging the values into the formula:

`$\[C(6 + 2 - 1, 2) = ((6 + 2 - 1)!)/(2! (6 - 1)!) = (7!)/(2! 5!)\]`

Calculating the value:

`$\[C(7, 2) = (7 * 6)/(2 * 1) = 21\]$`

So, the number of different double-scoop combinations possible is
`\(\boxed{21}\)`.